Set Theory Aritra Hazra Department of Computer Science and - - PowerPoint PPT Presentation

Set Theory

Aritra Hazra

Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, Paschim Medinipur, West Bengal, India - 721302. Email: aritrah@cse.iitkgp.ac.in

Autumn 2020

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 1 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10})

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10}) Membership: Element belonging to (or a member of) a set (Ex: 25, 64 ∈ S and 50, 72 ∈ S)

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10}) Membership: Element belonging to (or a member of) a set (Ex: 25, 64 ∈ S and 50, 72 ∈ S) Cardinality: Number of elements in a set (Ex: |S| = 9)

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10}) Membership: Element belonging to (or a member of) a set (Ex: 25, 64 ∈ S and 50, 72 ∈ S) Cardinality: Number of elements in a set (Ex: |S| = 9) Finite Set: Set having finite cardinality (Ex: The set, S) Infinite Set: Set having infinite (∞) cardinality (Ex: T = {1, 2, 4, 8, 16, . . .} = {2y | y is integer and y ≥ 0})

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10}) Membership: Element belonging to (or a member of) a set (Ex: 25, 64 ∈ S and 50, 72 ∈ S) Cardinality: Number of elements in a set (Ex: |S| = 9) Finite Set: Set having finite cardinality (Ex: The set, S) Infinite Set: Set having infinite (∞) cardinality (Ex: T = {1, 2, 4, 8, 16, . . .} = {2y | y is integer and y ≥ 0}) Subset: A set (A) is a subset of another set (B) iff each element of A is also a member of B. Formally, A ⊆ B iff ∀x [x ∈ A ⇒ x ∈ B].

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10}) Membership: Element belonging to (or a member of) a set (Ex: 25, 64 ∈ S and 50, 72 ∈ S) Cardinality: Number of elements in a set (Ex: |S| = 9) Finite Set: Set having finite cardinality (Ex: The set, S) Infinite Set: Set having infinite (∞) cardinality (Ex: T = {1, 2, 4, 8, 16, . . .} = {2y | y is integer and y ≥ 0}) Subset: A set (A) is a subset of another set (B) iff each element of A is also a member of B. Formally, A ⊆ B iff ∀x [x ∈ A ⇒ x ∈ B]. Hence, A ⊆ B iff ¬∀x [x ∈ A ⇒ x ∈ B] ≡ ∃x [x ∈ A ∧ x ∈ B].

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10}) Membership: Element belonging to (or a member of) a set (Ex: 25, 64 ∈ S and 50, 72 ∈ S) Cardinality: Number of elements in a set (Ex: |S| = 9) Finite Set: Set having finite cardinality (Ex: The set, S) Infinite Set: Set having infinite (∞) cardinality (Ex: T = {1, 2, 4, 8, 16, . . .} = {2y | y is integer and y ≥ 0}) Subset: A set (A) is a subset of another set (B) iff each element of A is also a member of B. Formally, A ⊆ B iff ∀x [x ∈ A ⇒ x ∈ B]. Hence, A ⊆ B iff ¬∀x [x ∈ A ⇒ x ∈ B] ≡ ∃x [x ∈ A ∧ x ∈ B]. (Ex: Let R = {z | z is composite integer and 2 ≤ z ≤ 100}, so S ⊆ R)

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10}) Membership: Element belonging to (or a member of) a set (Ex: 25, 64 ∈ S and 50, 72 ∈ S) Cardinality: Number of elements in a set (Ex: |S| = 9) Finite Set: Set having finite cardinality (Ex: The set, S) Infinite Set: Set having infinite (∞) cardinality (Ex: T = {1, 2, 4, 8, 16, . . .} = {2y | y is integer and y ≥ 0}) Subset: A set (A) is a subset of another set (B) iff each element of A is also a member of B. Formally, A ⊆ B iff ∀x [x ∈ A ⇒ x ∈ B]. Hence, A ⊆ B iff ¬∀x [x ∈ A ⇒ x ∈ B] ≡ ∃x [x ∈ A ∧ x ∈ B]. (Ex: Let R = {z | z is composite integer and 2 ≤ z ≤ 100}, so S ⊆ R) Equal Sets: A = B iff [(A ⊆ B) ∧ (B ⊆ A)] ≡ ∀x [x ∈ A ⇔ x ∈ B]

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10}) Membership: Element belonging to (or a member of) a set (Ex: 25, 64 ∈ S and 50, 72 ∈ S) Cardinality: Number of elements in a set (Ex: |S| = 9) Finite Set: Set having finite cardinality (Ex: The set, S) Infinite Set: Set having infinite (∞) cardinality (Ex: T = {1, 2, 4, 8, 16, . . .} = {2y | y is integer and y ≥ 0}) Subset: A set (A) is a subset of another set (B) iff each element of A is also a member of B. Formally, A ⊆ B iff ∀x [x ∈ A ⇒ x ∈ B]. Hence, A ⊆ B iff ¬∀x [x ∈ A ⇒ x ∈ B] ≡ ∃x [x ∈ A ∧ x ∈ B]. (Ex: Let R = {z | z is composite integer and 2 ≤ z ≤ 100}, so S ⊆ R) Equal Sets: A = B iff [(A ⊆ B) ∧ (B ⊆ A)] ≡ ∀x [x ∈ A ⇔ x ∈ B] Proper Subset: A ⊂ B iff [∀x (x ∈ A ⇒ x ∈ B) ∧ ∃y (y ∈ B ∧ y ∈ A)]

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10}) Membership: Element belonging to (or a member of) a set (Ex: 25, 64 ∈ S and 50, 72 ∈ S) Cardinality: Number of elements in a set (Ex: |S| = 9) Finite Set: Set having finite cardinality (Ex: The set, S) Infinite Set: Set having infinite (∞) cardinality (Ex: T = {1, 2, 4, 8, 16, . . .} = {2y | y is integer and y ≥ 0}) Subset: A set (A) is a subset of another set (B) iff each element of A is also a member of B. Formally, A ⊆ B iff ∀x [x ∈ A ⇒ x ∈ B]. Hence, A ⊆ B iff ¬∀x [x ∈ A ⇒ x ∈ B] ≡ ∃x [x ∈ A ∧ x ∈ B]. (Ex: Let R = {z | z is composite integer and 2 ≤ z ≤ 100}, so S ⊆ R) Equal Sets: A = B iff [(A ⊆ B) ∧ (B ⊆ A)] ≡ ∀x [x ∈ A ⇔ x ∈ B] Proper Subset: A ⊂ B iff [∀x (x ∈ A ⇒ x ∈ B) ∧ ∃y (y ∈ B ∧ y ∈ A)] Null Set: Set containing NO element, denoted using φ or {} (Ex: Q = {z | x + y = z and all x, y, z are odd} = φ)

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Sets and Subsets: Definitions and Properties

Set: Well-defined collection of distinct objects (Ex: S = {4, 9, 16 . . . , 81, 100} = {x2 | x is integer and 1 < x ≤ 10}) Membership: Element belonging to (or a member of) a set (Ex: 25, 64 ∈ S and 50, 72 ∈ S) Cardinality: Number of elements in a set (Ex: |S| = 9) Finite Set: Set having finite cardinality (Ex: The set, S) Infinite Set: Set having infinite (∞) cardinality (Ex: T = {1, 2, 4, 8, 16, . . .} = {2y | y is integer and y ≥ 0}) Subset: A set (A) is a subset of another set (B) iff each element of A is also a member of B. Formally, A ⊆ B iff ∀x [x ∈ A ⇒ x ∈ B]. Hence, A ⊆ B iff ¬∀x [x ∈ A ⇒ x ∈ B] ≡ ∃x [x ∈ A ∧ x ∈ B]. (Ex: Let R = {z | z is composite integer and 2 ≤ z ≤ 100}, so S ⊆ R) Equal Sets: A = B iff [(A ⊆ B) ∧ (B ⊆ A)] ≡ ∀x [x ∈ A ⇔ x ∈ B] Proper Subset: A ⊂ B iff [∀x (x ∈ A ⇒ x ∈ B) ∧ ∃y (y ∈ B ∧ y ∈ A)] Null Set: Set containing NO element, denoted using φ or {} (Ex: Q = {z | x + y = z and all x, y, z are odd} = φ) Note: |φ| = 0, but φ = {0} and φ = {φ} (since, |{0}| = |{φ}| = 1)

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 11

Power Set and Set Properties

Power Set: Set of all possible subsets of a set (A), denoted as P(A) or 2A (Ex: Let A = {1, 2, 3}, Thus, P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}})

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 11

Power Set and Set Properties

Power Set: Set of all possible subsets of a set (A), denoted as P(A) or 2A (Ex: Let A = {1, 2, 3}, Thus, P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}) Cardinality: |P(A)| = 2|A| (Why?)

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 11

Power Set and Set Properties

Power Set: Set of all possible subsets of a set (A), denoted as P(A) or 2A (Ex: Let A = {1, 2, 3}, Thus, P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}) Cardinality: |P(A)| = 2|A| (Why?) Proof: Let |A| = n. There are n

k

• subsets of size k possible (for any k,

0 ≤ k ≤ n). So, the total number of subsets = n

i=0

n

k

• = 2n.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 11

Power Set and Set Properties

Power Set: Set of all possible subsets of a set (A), denoted as P(A) or 2A (Ex: Let A = {1, 2, 3}, Thus, P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}) Cardinality: |P(A)| = 2|A| (Why?) Proof: Let |A| = n. There are n

k

• subsets of size k possible (for any k,

0 ≤ k ≤ n). So, the total number of subsets = n

i=0

n

k

• = 2n.

Properties: For sets A, B, C, we have the following:

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 11

Power Set and Set Properties

Power Set: Set of all possible subsets of a set (A), denoted as P(A) or 2A (Ex: Let A = {1, 2, 3}, Thus, P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}) Cardinality: |P(A)| = 2|A| (Why?) Proof: Let |A| = n. There are n

k

• subsets of size k possible (for any k,

0 ≤ k ≤ n). So, the total number of subsets = n

i=0

n

k

• = 2n.

Properties: For sets A, B, C, we have the following: A ⊂ B ⇒ A ⊆ B, but A ⊆ B ⇒ A ⊂ B.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 11

Power Set and Set Properties

Power Set: Set of all possible subsets of a set (A), denoted as P(A) or 2A (Ex: Let A = {1, 2, 3}, Thus, P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}) Cardinality: |P(A)| = 2|A| (Why?) Proof: Let |A| = n. There are n

k

• subsets of size k possible (for any k,

0 ≤ k ≤ n). So, the total number of subsets = n

i=0

n

k

• = 2n.

Properties: For sets A, B, C, we have the following: A ⊂ B ⇒ A ⊆ B, but A ⊆ B ⇒ A ⊂ B. (A ⊂ B) if and only if [(A ⊆ B) ∧ (A = B)].

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 11

Power Set and Set Properties

Power Set: Set of all possible subsets of a set (A), denoted as P(A) or 2A (Ex: Let A = {1, 2, 3}, Thus, P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}) Cardinality: |P(A)| = 2|A| (Why?) Proof: Let |A| = n. There are n

k

• subsets of size k possible (for any k,

0 ≤ k ≤ n). So, the total number of subsets = n

i=0

n

k

• = 2n.

Properties: For sets A, B, C, we have the following: A ⊂ B ⇒ A ⊆ B, but A ⊆ B ⇒ A ⊂ B. (A ⊂ B) if and only if [(A ⊆ B) ∧ (A = B)]. (A = B) if and only if (A ⊆ B) ∨ (B ⊆ A).

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 11

Power Set and Set Properties

Power Set: Set of all possible subsets of a set (A), denoted as P(A) or 2A (Ex: Let A = {1, 2, 3}, Thus, P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}) Cardinality: |P(A)| = 2|A| (Why?) Proof: Let |A| = n. There are n

k

• subsets of size k possible (for any k,

0 ≤ k ≤ n). So, the total number of subsets = n

i=0

n

k

• = 2n.

Properties: For sets A, B, C, we have the following: A ⊂ B ⇒ A ⊆ B, but A ⊆ B ⇒ A ⊂ B. (A ⊂ B) if and only if [(A ⊆ B) ∧ (A = B)]. (A = B) if and only if (A ⊆ B) ∨ (B ⊆ A). φ ⊆ A. If A = φ, then φ ⊂ A. A ⊆ A and A ∈ P(A).

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 11

Power Set and Set Properties

Power Set: Set of all possible subsets of a set (A), denoted as P(A) or 2A (Ex: Let A = {1, 2, 3}, Thus, P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}) Cardinality: |P(A)| = 2|A| (Why?) Proof: Let |A| = n. There are n

k

• subsets of size k possible (for any k,

0 ≤ k ≤ n). So, the total number of subsets = n

i=0

n

k

• = 2n.

Properties: For sets A, B, C, we have the following: A ⊂ B ⇒ A ⊆ B, but A ⊆ B ⇒ A ⊂ B. (A ⊂ B) if and only if [(A ⊆ B) ∧ (A = B)]. (A = B) if and only if (A ⊆ B) ∨ (B ⊆ A). φ ⊆ A. If A = φ, then φ ⊂ A. A ⊆ A and A ∈ P(A). If (A ⊆ B), then |A| ≤ |B|. If (A ⊂ B), then |A| < |B|. If (A = B), then |A| = |B|.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 11

Power Set and Set Properties

Power Set: Set of all possible subsets of a set (A), denoted as P(A) or 2A (Ex: Let A = {1, 2, 3}, Thus, P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}) Cardinality: |P(A)| = 2|A| (Why?) Proof: Let |A| = n. There are n

k

• subsets of size k possible (for any k,

0 ≤ k ≤ n). So, the total number of subsets = n

i=0

n

k

• = 2n.

Properties: For sets A, B, C, we have the following: A ⊂ B ⇒ A ⊆ B, but A ⊆ B ⇒ A ⊂ B. (A ⊂ B) if and only if [(A ⊆ B) ∧ (A = B)]. (A = B) if and only if (A ⊆ B) ∨ (B ⊆ A). φ ⊆ A. If A = φ, then φ ⊂ A. A ⊆ A and A ∈ P(A). If (A ⊆ B), then |A| ≤ |B|. If (A ⊂ B), then |A| < |B|. If (A = B), then |A| = |B|. If (A ⊆ B) and (B ⊆ C), then (A ⊆ C). If (A ⊂ B) and (B ⊂ C), then (A ⊂ C). If (A ⊆ B) and (B ⊂ C), then (A ⊂ C). If (A ⊂ B) and (B ⊆ C), then (A ⊂ C).

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 11

Frequently-Used Set Examples and Notations

Popular Set Examples: N = Set of Non-negative natural numbers = {0, 1, 2, . . .} Z = Set of Integers = {. . . , −2, −1, 0, 1, 2, . . .} Z+ = Set of Positive Integers = {x ∈ Z | x > 0} Q = Set of Rational Numbers = { a

b | a, b ∈ Z, b = 0}

Q+ = Set of Positive Rational Numbers = {r ∈ Q | r > 0} Q∗ = Set of Non-zero Rational Numbers = {r ∈ Q | r = 0} R = Set of Real Numbers R+ = Set of Positive Real Numbers R∗ = Set of Non-zero Real Numbers C = Set of Complex Numbers = {a + ib | a, b ∈ R, i2 = −1} C∗ = Set of Non-zero Complex Numbers = {c ∈ C | c = 0}

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 4 / 11

Frequently-Used Set Examples and Notations

Popular Set Examples: N = Set of Non-negative natural numbers = {0, 1, 2, . . .} Z = Set of Integers = {. . . , −2, −1, 0, 1, 2, . . .} Z+ = Set of Positive Integers = {x ∈ Z | x > 0} Q = Set of Rational Numbers = { a

b | a, b ∈ Z, b = 0}

Q+ = Set of Positive Rational Numbers = {r ∈ Q | r > 0} Q∗ = Set of Non-zero Rational Numbers = {r ∈ Q | r = 0} R = Set of Real Numbers R+ = Set of Positive Real Numbers R∗ = Set of Non-zero Real Numbers C = Set of Complex Numbers = {a + ib | a, b ∈ R, i2 = −1} C∗ = Set of Non-zero Complex Numbers = {c ∈ C | c = 0} Frequently-Used Notations: For each n ∈ Z+, Zn = {0, 1, 2, . . . , n − 1} For real numbers, a, b with a < b, we define intervals as follows: (Closed) [a, b] = {x | a ≤ x ≤ b} (Open) (a, b) = {x | a < x < b} (Half-Open) (a, b] = {x | a < x ≤ b} and [a, b) = {x | a ≤ x < b}

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 4 / 11

Counting using Set Theory

Prove that, n

r+1

• = n−r

r+1

n

r

• Aritra Hazra (CSE, IITKGP)

CS21001 : Discrete Structures Autumn 2020 5 / 11

Counting using Set Theory

Prove that, n

r+1

• = n−r

r+1

n

r

• Counting: Total number of (r + 1)-element subsets, formed from all r-element

subsets by adding an element from (n − r) remaining elements, is, m = (n − r) n

r

• .

Ex: Let n = 4 and S = {1, 2, 3, 4}. All 2-element subsets are, A1 = {1, 2}, A2 = {1, 3}, A3 = {1, 4}, A4 = {2, 3}, A5 = {2, 4}, A6 = {3, 4}. From each Ais, a 3-element subset can be formed in two ways. So, total possibilities = 2 × 4

2

• = 12.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 5 / 11

Counting using Set Theory

Prove that, n

r+1

• = n−r

r+1

n

r

• Counting: Total number of (r + 1)-element subsets, formed from all r-element

subsets by adding an element from (n − r) remaining elements, is, m = (n − r) n

r

• .

Ex: Let n = 4 and S = {1, 2, 3, 4}. All 2-element subsets are, A1 = {1, 2}, A2 = {1, 3}, A3 = {1, 4}, A4 = {2, 3}, A5 = {2, 4}, A6 = {3, 4}. From each Ais, a 3-element subset can be formed in two ways. So, total possibilities = 2 × 4

2

• = 12.

Repetition: Each (r + 1) element subset can be formed from (r + 1) different r-element subsets. So, the total choice reduces to, n

r+1

• =

m r+1 = n−r r+1

n

r

• .

Ex: 3-element subset {1, 2, 3} can formed from A1, A2, A4 by adding an element to

• each. So, reduced number of possibilities = 12

3 = 4 =

4

3

• Aritra Hazra (CSE, IITKGP)

CS21001 : Discrete Structures Autumn 2020 5 / 11

Counting using Set Theory

Prove that, n

r+1

• = n−r

r+1

n

r

• Counting: Total number of (r + 1)-element subsets, formed from all r-element

subsets by adding an element from (n − r) remaining elements, is, m = (n − r) n

r

• .

Ex: Let n = 4 and S = {1, 2, 3, 4}. All 2-element subsets are, A1 = {1, 2}, A2 = {1, 3}, A3 = {1, 4}, A4 = {2, 3}, A5 = {2, 4}, A6 = {3, 4}. From each Ais, a 3-element subset can be formed in two ways. So, total possibilities = 2 × 4

2

• = 12.

Repetition: Each (r + 1) element subset can be formed from (r + 1) different r-element subsets. So, the total choice reduces to, n

r+1

• =

m r+1 = n−r r+1

n

r

• .

Ex: 3-element subset {1, 2, 3} can formed from A1, A2, A4 by adding an element to

• each. So, reduced number of possibilities = 12

3 = 4 =

4

3

• Prove that,

r

r

• +

r+1

r

• + · · · +

n−1

r

• +

n

r

• =

n+1

r+1

• Aritra Hazra (CSE, IITKGP)

CS21001 : Discrete Structures Autumn 2020 5 / 11

Counting using Set Theory

Prove that, n

r+1

• = n−r

r+1

n

r

• Counting: Total number of (r + 1)-element subsets, formed from all r-element

subsets by adding an element from (n − r) remaining elements, is, m = (n − r) n

r

• .

Ex: Let n = 4 and S = {1, 2, 3, 4}. All 2-element subsets are, A1 = {1, 2}, A2 = {1, 3}, A3 = {1, 4}, A4 = {2, 3}, A5 = {2, 4}, A6 = {3, 4}. From each Ais, a 3-element subset can be formed in two ways. So, total possibilities = 2 × 4

2

• = 12.

Repetition: Each (r + 1) element subset can be formed from (r + 1) different r-element subsets. So, the total choice reduces to, n

r+1

• =

m r+1 = n−r r+1

n

r

• .

Ex: 3-element subset {1, 2, 3} can formed from A1, A2, A4 by adding an element to

• each. So, reduced number of possibilities = 12

3 = 4 =

4

3

• Prove that,

r

r

• +

r+1

r

• + · · · +

n−1

r

• +

n

r

• =

n+1

r+1

• Let the (n + 1)-element set be = {1, 2, . . ., n, n + 1} From (n + 1)-element set,

choosing (r + 1)-element subsets with smallest element i can be done in n+1−i

r

• ways. So, all such possible choice leads to, (n+1)−(r+1)

i=1

n+1−i

r

• =

n+1

r+1

• ,

implying the proof.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 5 / 11

Counting using Set Theory

Prove that, n

i=0 i

n

i

• = n.2n−1

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 6 / 11

Counting using Set Theory

Prove that, n

i=0 i

n

i

• = n.2n−1

From an n-element set, Size of a subset with i elements + Size of its complement subset = i + (n − i) = n and there are n

i

• number of these each.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 6 / 11

Counting using Set Theory

Prove that, n

i=0 i

n

i

• = n.2n−1

From an n-element set, Size of a subset with i elements + Size of its complement subset = i + (n − i) = n and there are n

i

• number of these each.

Therefore, 2 n

i=0 i

n

i

• = n n

i=0

n

i

• = n.2n, implying the proof.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 6 / 11

Counting using Set Theory

Prove that, n

i=0 i

n

i

• = n.2n−1

From an n-element set, Size of a subset with i elements + Size of its complement subset = i + (n − i) = n and there are n

i

• number of these each.

Therefore, 2 n

i=0 i

n

i

• = n n

i=0

n

i

• = n.2n, implying the proof.

Prove that, Number of Summands of n is 2n−1.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 6 / 11

Counting using Set Theory

Prove that, n

i=0 i

n

i

• = n.2n−1

From an n-element set, Size of a subset with i elements + Size of its complement subset = i + (n − i) = n and there are n

i

• number of these each.

Therefore, 2 n

i=0 i

n

i

• = n n

i=0

n

i

• = n.2n, implying the proof.

Prove that, Number of Summands of n is 2n−1.

Consider, n = 4. Summand Subset Correspondence 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 φ 2 + 1 + 1 = (1+1) + 1 + 1 {1} 1 + 2 + 1 = 1 + (1+1) + 1 {2} 1 + 1 + 2 = 1 + 1 + (1+1) {3} 3 + 1 = (1+1+1) + 1 {1, 2} 2 + 2 = (1+1) + (1+1) {1, 3} 1 + 3 = 1 + (1+1+1) {2, 3} 4 = (1+1+1+1) {1, 2, 3}

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 6 / 11

Counting using Set Theory

Prove that, n

i=0 i

n

i

• = n.2n−1

From an n-element set, Size of a subset with i elements + Size of its complement subset = i + (n − i) = n and there are n

i

• number of these each.

Therefore, 2 n

i=0 i

n

i

• = n n

i=0

n

i

• = n.2n, implying the proof.

Prove that, Number of Summands of n is 2n−1.

Consider, n = 4. Summand Subset Correspondence 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 φ 2 + 1 + 1 = (1+1) + 1 + 1 {1} 1 + 2 + 1 = 1 + (1+1) + 1 {2} 1 + 1 + 2 = 1 + 1 + (1+1) {3} 3 + 1 = (1+1+1) + 1 {1, 2} 2 + 2 = (1+1) + (1+1) {1, 3} 1 + 3 = 1 + (1+1+1) {2, 3} 4 = (1+1+1+1) {1, 2, 3} ∴ Number of summands of n = Number of subsets of an (n − 1)-element set = 2n−1.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 6 / 11

Set Operations

For two sets, A, B ∈ U (universal set), the following operations are defined: (Ex: Let, A = {1, 2, 3} and B = {2, 3, 4}) Union: A ∪ B = {x | x ∈ A ∨ x ∈ B} (Ex: A ∪ B = {1, 2, 3, 4})

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 7 / 11

Set Operations

For two sets, A, B ∈ U (universal set), the following operations are defined: (Ex: Let, A = {1, 2, 3} and B = {2, 3, 4}) Union: A ∪ B = {x | x ∈ A ∨ x ∈ B} (Ex: A ∪ B = {1, 2, 3, 4}) Intersection: A ∩ B = {x | x ∈ A ∧ x ∈ B} (Ex: A ∩ B = {2, 3})

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 7 / 11

Set Operations

For two sets, A, B ∈ U (universal set), the following operations are defined: (Ex: Let, A = {1, 2, 3} and B = {2, 3, 4}) Union: A ∪ B = {x | x ∈ A ∨ x ∈ B} (Ex: A ∪ B = {1, 2, 3, 4}) Intersection: A ∩ B = {x | x ∈ A ∧ x ∈ B} (Ex: A ∩ B = {2, 3}) Complement: A = {x | x ∈ U ∧ x ∈ A}

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 7 / 11

Set Operations

For two sets, A, B ∈ U (universal set), the following operations are defined: (Ex: Let, A = {1, 2, 3} and B = {2, 3, 4}) Union: A ∪ B = {x | x ∈ A ∨ x ∈ B} (Ex: A ∪ B = {1, 2, 3, 4}) Intersection: A ∩ B = {x | x ∈ A ∧ x ∈ B} (Ex: A ∩ B = {2, 3}) Complement: A = {x | x ∈ U ∧ x ∈ A} Relative Complement: A − B = {x | x ∈ A ∧ x ∈ B} = A ∩ B (Ex: A − B = {1})

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 7 / 11

Set Operations

For two sets, A, B ∈ U (universal set), the following operations are defined: (Ex: Let, A = {1, 2, 3} and B = {2, 3, 4}) Union: A ∪ B = {x | x ∈ A ∨ x ∈ B} (Ex: A ∪ B = {1, 2, 3, 4}) Intersection: A ∩ B = {x | x ∈ A ∧ x ∈ B} (Ex: A ∩ B = {2, 3}) Complement: A = {x | x ∈ U ∧ x ∈ A} Relative Complement: A − B = {x | x ∈ A ∧ x ∈ B} = A ∩ B (Ex: A − B = {1}) Symmetric Difference: A ∆ B = {x | (x ∈ A ∨ x ∈ B) ∧ x ∈ A ∩ B} = {x | x ∈ A ∪ B ∧ x ∈ A ∩ B} = (A ∪ B) − (A ∩ B) = {x | x ∈ A ∩ B ∧ x ∈ A ∩ B} = (A ∩ B) ∪ (A ∩ B) = (A − B) ∪ (B − A) = B ∆ A (Ex: A ∆ B = {1, 4})

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 7 / 11

Set Operations

For two sets, A, B ∈ U (universal set), the following operations are defined: (Ex: Let, A = {1, 2, 3} and B = {2, 3, 4}) Union: A ∪ B = {x | x ∈ A ∨ x ∈ B} (Ex: A ∪ B = {1, 2, 3, 4}) Intersection: A ∩ B = {x | x ∈ A ∧ x ∈ B} (Ex: A ∩ B = {2, 3}) Complement: A = {x | x ∈ U ∧ x ∈ A} Relative Complement: A − B = {x | x ∈ A ∧ x ∈ B} = A ∩ B (Ex: A − B = {1}) Symmetric Difference: A ∆ B = {x | (x ∈ A ∨ x ∈ B) ∧ x ∈ A ∩ B} = {x | x ∈ A ∪ B ∧ x ∈ A ∩ B} = (A ∪ B) − (A ∩ B) = {x | x ∈ A ∩ B ∧ x ∈ A ∩ B} = (A ∩ B) ∪ (A ∩ B) = (A − B) ∪ (B − A) = B ∆ A (Ex: A ∆ B = {1, 4}) Mutual Disjoint: Sets, A and B, are mutually disjoint (or disjoint), when A ∩ B = φ. In such a case, A ∆ B = A ∪ B, A ∩ B = A and A ∩ B = B.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 7 / 11

Set Operations

For two sets, A, B ∈ U (universal set), the following operations are defined: (Ex: Let, A = {1, 2, 3} and B = {2, 3, 4}) Union: A ∪ B = {x | x ∈ A ∨ x ∈ B} (Ex: A ∪ B = {1, 2, 3, 4}) Intersection: A ∩ B = {x | x ∈ A ∧ x ∈ B} (Ex: A ∩ B = {2, 3}) Complement: A = {x | x ∈ U ∧ x ∈ A} Relative Complement: A − B = {x | x ∈ A ∧ x ∈ B} = A ∩ B (Ex: A − B = {1}) Symmetric Difference: A ∆ B = {x | (x ∈ A ∨ x ∈ B) ∧ x ∈ A ∩ B} = {x | x ∈ A ∪ B ∧ x ∈ A ∩ B} = (A ∪ B) − (A ∩ B) = {x | x ∈ A ∩ B ∧ x ∈ A ∩ B} = (A ∩ B) ∪ (A ∩ B) = (A − B) ∪ (B − A) = B ∆ A (Ex: A ∆ B = {1, 4}) Mutual Disjoint: Sets, A and B, are mutually disjoint (or disjoint), when A ∩ B = φ. In such a case, A ∆ B = A ∪ B, A ∩ B = A and A ∩ B = B. The following statements are equivalent: (Proof Left as an Exercise!) (a) A ⊆ B, (b) A ∪ B = B, (c) A ∩ B = A, (d) B ⊆ A

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 7 / 11

Laws of Set Theory

For three sets, A, B, C ∈ U, the rules given as follows: Name of the Law Mathematical Expressions Double Complement: A = A DeMorgan’s Laws: A ∪ B = A ∩ B, A ∩ B = A ∪ B Commutative Laws: A ∪ B = B ∪ A, A ∩ B = B ∩ A Associative Laws: A ∪ (B ∪ C) = (A ∪ B) ∪ C, A ∩ (B ∩ C) = (A ∩ B) ∩ C Distributive Laws:

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Idempotent Laws: A ∪ A = A, A ∩ A = A Identity Laws: A ∪ φ = A, A ∩ U = A Inverse Laws: A ∪ A = U, A ∩ A = φ Domination Laws: A ∪ U = U, A ∩ φ = φ Absorption Laws: A ∪ (A ∩ B) = A, A ∩ (A ∪ B) = A

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 8 / 11

Laws of Set Theory

For three sets, A, B, C ∈ U, the rules given as follows: Name of the Law Mathematical Expressions Double Complement: A = A DeMorgan’s Laws: A ∪ B = A ∩ B, A ∩ B = A ∪ B Commutative Laws: A ∪ B = B ∪ A, A ∩ B = B ∩ A Associative Laws: A ∪ (B ∪ C) = (A ∪ B) ∪ C, A ∩ (B ∩ C) = (A ∩ B) ∩ C Distributive Laws:

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Idempotent Laws: A ∪ A = A, A ∩ A = A Identity Laws: A ∪ φ = A, A ∩ U = A Inverse Laws: A ∪ A = U, A ∩ A = φ Domination Laws: A ∪ U = U, A ∩ φ = φ Absorption Laws: A ∪ (A ∩ B) = A, A ∩ (A ∪ B) = A

An Example Proof Sketch: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

x ∈ A ∪ (B ∩ C) ⇔ (x ∈ A) ∨ (x ∈ B ∩ C) ⇔ (x ∈ A) ∨ ((x ∈ B) ∧ (x ∈ C)) ⇔ ((x ∈ A) ∨ (x ∈ B)) ∧ ((x ∈ A) ∨ (x ∈ C)) ⇔ (x ∈ A ∪ B) ∧ (x ∈ A ∪ C) ⇔ x ∈ (A ∪ B) ∩ (A ∪ C)

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 8 / 11

Some Derived Laws and Observations

A1 = A1 ∪ (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) ∪ · · · (∀i, Ai ∈ U)

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 9 / 11

Some Derived Laws and Observations

A1 = A1 ∪ (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) ∪ · · · (∀i, Ai ∈ U) Proof: A1 ∪ (A1 ∩ A2) = A1, (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) = (A1 ∩ A2), (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) = (A1 ∩ A2 ∩ A3), and so on ...

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 9 / 11

Some Derived Laws and Observations

A1 = A1 ∪ (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) ∪ · · · (∀i, Ai ∈ U) Proof: A1 ∪ (A1 ∩ A2) = A1, (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) = (A1 ∩ A2), (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) = (A1 ∩ A2 ∩ A3), and so on ... Similarly, A1 = A1 ∩ (A1 ∪ A2) ∩ (A1 ∪ A2 ∪ A3) ∩ (A1 ∪ A2 ∪ A3 ∪ A4) ∩ · · ·

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 9 / 11

Some Derived Laws and Observations

A1 = A1 ∪ (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) ∪ · · · (∀i, Ai ∈ U) Proof: A1 ∪ (A1 ∩ A2) = A1, (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) = (A1 ∩ A2), (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) = (A1 ∩ A2 ∩ A3), and so on ... Similarly, A1 = A1 ∩ (A1 ∪ A2) ∩ (A1 ∪ A2 ∪ A3) ∩ (A1 ∪ A2 ∪ A3 ∪ A4) ∩ · · · A ∆ B = A ∆ B = A ∆ B

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 9 / 11

Some Derived Laws and Observations

A1 = A1 ∪ (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) ∪ · · · (∀i, Ai ∈ U) Proof: A1 ∪ (A1 ∩ A2) = A1, (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) = (A1 ∩ A2), (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) = (A1 ∩ A2 ∩ A3), and so on ... Similarly, A1 = A1 ∩ (A1 ∪ A2) ∩ (A1 ∪ A2 ∪ A3) ∩ (A1 ∪ A2 ∪ A3 ∪ A4) ∩ · · · A ∆ B = A ∆ B = A ∆ B Proof: As, A ∆ B = (A ∪ B) − (A ∩ B) and A ∆ B = (A ∩ B) ∪ (A ∩ B), so A ∆ B = (A ∩ B) ∪ (A ∩ B) = (A ∪ B) ∩ (A ∩ B) = (A ∪ B) − (A ∩ B) = A ∆ B and A ∆ B = (A ∩ B) ∪ (A ∩ B) = (A ∪ B) ∩ (A ∩ B) = (A ∪ B) − (A ∩ B) = A ∆ B

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 9 / 11

Some Derived Laws and Observations

A1 = A1 ∪ (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) ∪ · · · (∀i, Ai ∈ U) Proof: A1 ∪ (A1 ∩ A2) = A1, (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) = (A1 ∩ A2), (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) = (A1 ∩ A2 ∩ A3), and so on ... Similarly, A1 = A1 ∩ (A1 ∪ A2) ∩ (A1 ∪ A2 ∪ A3) ∩ (A1 ∪ A2 ∪ A3 ∪ A4) ∩ · · · A ∆ B = A ∆ B = A ∆ B Proof: As, A ∆ B = (A ∪ B) − (A ∩ B) and A ∆ B = (A ∩ B) ∪ (A ∩ B), so A ∆ B = (A ∩ B) ∪ (A ∩ B) = (A ∪ B) ∩ (A ∩ B) = (A ∪ B) − (A ∩ B) = A ∆ B and A ∆ B = (A ∩ B) ∪ (A ∩ B) = (A ∪ B) ∩ (A ∩ B) = (A ∪ B) − (A ∩ B) = A ∆ B A − (B ∪ C) = (A − B) ∩ (A − C) (A ∪ B) − C = (A − C) ∪ (B − C) A − (B ∩ C) = (A − B) ∪ (A − C) (A ∩ B) − C = (A − C) ∩ (B − C) (A ∩ B) − (A ∩ C) = A ∩ (B − C)

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 9 / 11

Some Derived Laws and Observations

A1 = A1 ∪ (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) ∪ · · · (∀i, Ai ∈ U) Proof: A1 ∪ (A1 ∩ A2) = A1, (A1 ∩ A2) ∪ (A1 ∩ A2 ∩ A3) = (A1 ∩ A2), (A1 ∩ A2 ∩ A3) ∪ (A1 ∩ A2 ∩ A3 ∩ A4) = (A1 ∩ A2 ∩ A3), and so on ... Similarly, A1 = A1 ∩ (A1 ∪ A2) ∩ (A1 ∪ A2 ∪ A3) ∩ (A1 ∪ A2 ∪ A3 ∪ A4) ∩ · · · A ∆ B = A ∆ B = A ∆ B Proof: As, A ∆ B = (A ∪ B) − (A ∩ B) and A ∆ B = (A ∩ B) ∪ (A ∩ B), so A ∆ B = (A ∩ B) ∪ (A ∩ B) = (A ∪ B) ∩ (A ∩ B) = (A ∪ B) − (A ∩ B) = A ∆ B and A ∆ B = (A ∩ B) ∪ (A ∩ B) = (A ∪ B) ∩ (A ∩ B) = (A ∪ B) − (A ∩ B) = A ∆ B A − (B ∪ C) = (A − B) ∩ (A − C) (A ∪ B) − C = (A − C) ∪ (B − C) A − (B ∩ C) = (A − B) ∪ (A − C) (A ∩ B) − C = (A − C) ∩ (B − C) (A ∩ B) − (A ∩ C) = A ∩ (B − C) A ∆ B = A ∆ B = B ∆ A = B ∆ A A ∆ (B ∆ C) = (A ∆ B) ∆ C A ∩ (B ∆ C) = (A ∩ B) ∆ (A ∩ C)

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 9 / 11

Index Set and Partitions

Index Set

Definition: Let I = φ and ∀i ∈ I, let Ai ⊆ U (universal set). Then, I is called an index set, and each i ∈ I is an index.

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 10 / 11

Index Set and Partitions

Index Set

Definition: Let I = φ and ∀i ∈ I, let Ai ⊆ U (universal set). Then, I is called an index set, and each i ∈ I is an index. Set Operations: (Union)

i∈I Ai = {x | ∃i ∈ I, x ∈ Ai}

(Intersection)

i∈I Ai = {x | ∀i ∈ I, x ∈ Ai} Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 10 / 11

Index Set and Partitions

Index Set

Definition: Let I = φ and ∀i ∈ I, let Ai ⊆ U (universal set). Then, I is called an index set, and each i ∈ I is an index. Set Operations: (Union)

i∈I Ai = {x | ∃i ∈ I, x ∈ Ai}

(Intersection)

i∈I Ai = {x | ∀i ∈ I, x ∈ Ai}

Generalized DeMorgan’s Law:

i∈I Ai = i∈I Ai

and

• i∈I Ai =

i∈I Ai Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 10 / 11

Index Set and Partitions

Index Set

Definition: Let I = φ and ∀i ∈ I, let Ai ⊆ U (universal set). Then, I is called an index set, and each i ∈ I is an index. Set Operations: (Union)

i∈I Ai = {x | ∃i ∈ I, x ∈ Ai}

(Intersection)

i∈I Ai = {x | ∀i ∈ I, x ∈ Ai}

Generalized DeMorgan’s Law:

i∈I Ai = i∈I Ai

and

• i∈I Ai =

i∈I Ai

Partition of a Set

Definition: Let S be a non-empty set. A family of non-empty subsets, {Si | i ∈ I} (I being the index set) is said to form a partition of S if the following two condition holds:

• i∈I Si = S (Complete Set Cover), and

Si ∩ Sj = φ, ∀i, j ∈ I and i = j (Pairwise Disjoint).

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 10 / 11

Index Set and Partitions

Index Set

Definition: Let I = φ and ∀i ∈ I, let Ai ⊆ U (universal set). Then, I is called an index set, and each i ∈ I is an index. Set Operations: (Union)

i∈I Ai = {x | ∃i ∈ I, x ∈ Ai}

(Intersection)

i∈I Ai = {x | ∀i ∈ I, x ∈ Ai}

Generalized DeMorgan’s Law:

i∈I Ai = i∈I Ai

and

• i∈I Ai =

i∈I Ai

Partition of a Set

Definition: Let S be a non-empty set. A family of non-empty subsets, {Si | i ∈ I} (I being the index set) is said to form a partition of S if the following two condition holds:

• i∈I Si = S (Complete Set Cover), and

Si ∩ Sj = φ, ∀i, j ∈ I and i = j (Pairwise Disjoint). Example: Let Z0 = {3m | m is an integer} = {0, ±3, ±6, . . .}, Z1 = {3m + 1 | m is an integer} = {. . . , −8, −5, −2, +1, +4, +7, . . .} Z2 = {3m + 2 | m is an integer} = {. . . , −7, −4, −1, +2, +5, +8, . . .} Now, Z0 ∪ Z1 ∪ Z2 = Z and Z0 ∩ Z1 = Z1 ∩ Z2 = Z2 ∩ Z0 = φ

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 10 / 11

Thank You!

Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 11 / 11