Sets and Set Operations Dr. Philip C. Ritchey Set Notation Set: an - - PowerPoint PPT Presentation

Sets and Set Operations

• Dr. Philip C. Ritchey

CSCE 222 Discrete Structures for Computing

Set Notation

• Set: an unordered collection of objects (“members”, “elements”)
• 𝑏 ∈ 𝑇
• “𝑏 is a member of 𝑇”, “𝑏 is an element of 𝑇”
• 𝑏 ∉ 𝑇
• “𝑏 is not a member of 𝑇”, “𝑏 is not an element of 𝑇”
• Roster method
• S= 1,2,3,4
• Set Builder
• 𝑇 = 𝑦 ∣ 𝑦 𝑗𝑡 𝑏 𝑞𝑝𝑡𝑗𝑢𝑗𝑤𝑓 𝑗𝑜𝑢𝑓𝑕𝑓𝑠 𝑚𝑓𝑡𝑡 𝑢ℎ𝑓𝑜 5
• 𝑇 = 𝑦 ∣ 𝑦 ∈ ℤ+ ∧ 𝑦 < 5

Exercise

• Jumping Jacks!
• Kidding…
• List the members of the set:
• 𝑦 ∣ 𝑦 𝑗𝑡 𝑢ℎ𝑓 𝑡𝑟𝑣𝑏𝑠𝑓 𝑝𝑔 𝑏𝑜 𝑗𝑜𝑢𝑓𝑕𝑓𝑠 𝑏𝑜𝑒 𝑦 < 100
• 0,1,4,9,16,25,36,49,64,81
• 𝑦 ∣ 𝑦2 = 2
• − 2,

2

• Use set builder notation to describe the set:
• 0,3,6,9,12
• 3𝑦 ∣ 𝑦 𝑗𝑡 𝑏𝑜 𝑗𝑜𝑢𝑓𝑕𝑓𝑠 𝑏𝑜𝑒 0 ≤ 𝑦 ≤ 4
• −3, −2, −1,0,1,2,3
• 𝑦 ∣ 𝑦 𝑗𝑡 𝑏𝑜 𝑗𝑜𝑢𝑓𝑕𝑓𝑠 𝑏𝑜𝑒 𝑦 ≤ 3

Common Sets

• ℤ = … , −2, −1,0,1,2, … , the set of integers
• ℤ+, the positive integers
• ℤ−, the negative integers
• ℕ = [0, ]1,2,3, … , the set of natural numbers
• ℚ =

𝑞 𝑟 ∣ 𝑞, 𝑟 ∈ ℤ, 𝑟 ≠ 0 , the set of rational numbers

• ℝ, the set of real numbers
• ℝ+, the set of positive reals
• ℝ−, the set of negative reals
• ℂ, the set of complex numbers
• 𝑽: the universal set (set of discourse)
• ∅: the empty set,
• ∅ is a non-empty singleton set

Interval Notation

• Shortcuts for sets containing numbers
• [ , ] mean inclusive. Closed interval.
• ( , ) mean exclusive. Open interval.

𝑏, 𝑐 = 𝑦 ∣ 𝑏 ≤ 𝑦 ≤ 𝑐 𝑏, 𝑐 = 𝑦 ∣ 𝑏 < 𝑦 < 𝑐 𝑏, 𝑐 = 𝑦 ∣ 𝑏 ≤ 𝑦 < 𝑐 𝑏, 𝑐 = 𝑦 ∣ 𝑏 < 𝑦 ≤ 𝑐

Exercise

• Push Ups!
• Kidding…
• Use interval notation to describe the number line:
• −7, −1 ∪ 3,7
• Express 𝑦 ≠ 9 using:
• Set builder
• 𝑦 ∈ ℝ ∣ 𝑦 ≠ 9
• Interval Notation
• −∞, 9 ∪ 9, ∞

Set Equality and Subsets

• Set Equality, 𝐵 = 𝐶
• 𝐵 = 𝐶 ≔ ∀𝑦 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶
• 1,3,5 = 3,1,5
• 1,3,5 = 1,3,3,3,5,5,5,5,5 ?
• Subset, 𝐵 ⊆ 𝐶
• 𝐵 ⊆ 𝐶 ≔ ∀𝑦 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶
• 1,9 ⊆ 1,3,5,7,9
• 1,9 ⊆ 1,9
• Proper Subset, 𝐵 ⊂ 𝐶
• 𝐵 ⊂ 𝐶 ≔ 𝐵 ⊆ 𝐶 ∧ 𝐵 ≠ 𝐶
• 1,9 ⊂ 1,3,5,7,9
• 1,9 ⊄ 1,9
• ∅ ⊆ 𝐵, for any set 𝐵
• 𝐵 ⊆ 𝐵, for any set 𝐵
• How can we express equality in terms
• f subsets?
• 𝐵 = 𝐶 ↔ 𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵

Set Size and Power Sets

• Set size
• The number of distinct elements in the set
• 𝑇 = 𝑜, 𝑜 ≥ 0, 𝑜 is finite
• “cardinality”
• Power set
• The set of all subsets of a set
• ℘ 𝑇
• Ex: 𝑇 = 1,2,3
• ℘ 𝑇 = ∅, 1 , 2 , 3 , 1,2 , 1,3 , 2,3 , 1,2,3
• ℘ 𝑇

= 2 𝑇

Cartesian Products

• 𝐵 × 𝐶 =

𝑏, 𝑐 ∣ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶

• 𝐵1 × 𝐵2 × ⋯ × 𝐵𝑜 =

𝑏1, 𝑏2, … , 𝑏𝑜 ∣ 𝑏𝑗 ∈ 𝐵𝑗

• 𝑏1, 𝑏2, … , 𝑏𝑜 is an 𝑜-tuple
• 𝐵2 = 𝐵 × 𝐵
• 𝐵𝑜 =

𝑏1, 𝑏2, … , 𝑏𝑜 ∣ 𝑏𝑗 ∈ 𝐵

• 𝐵 × 𝐶 ≠ 𝐶 × 𝐵
• Exceptions: 𝐵 = ∅, 𝐶 = ∅, 𝐵 = 𝐶
• A subset of a Cartesian product is called a Relation from 𝐵 to 𝐶
• We will cover Relations soon.

Exercise

• Let 𝐵 be the set of all airlines and 𝐶 be the set of all US cities.
• What is the Cartesian product 𝐵 × 𝐶2?
• 𝑏, 𝑐, 𝑑 ∣ 𝑏 𝑗𝑡 𝑏𝑜 𝑏𝑗𝑠𝑚𝑗𝑜𝑓, 𝑐 𝑗𝑡 𝑏 𝑉𝑇 𝑑𝑗𝑢𝑧 𝑏𝑜𝑒 𝑑 𝑗𝑡 𝑏 𝑉𝑇 𝑑𝑗𝑢𝑧
• How can this Cartesian product be used?
• Each element is a route that an airline flies, from one city to another
• Could be used for route planning
• Does 𝐵 × 𝐶 × 𝐷 × 𝐸 = 𝐵 × 𝐶 × 𝐷 × 𝐸?
• No.

𝑏, 𝑐 , 𝑑, 𝑒 ≠ 𝑏, 𝑐, 𝑑 , 𝑒

Set Operations

• Set Union, 𝐵 ∪ 𝐶
• 𝐵 ∪ 𝐶 = 𝑦 ∣ 𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶
• 1,2,3 ∪ 3,4 = 1,2,3,4
• Set Intersection, 𝐵 ∩ 𝐶
• 𝐵 ∩ 𝐶 = 𝑦 ∣ 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶
• 1,2,3 ∩ 3,4 = 3
• If 𝐵 ∩ 𝐶 = ∅, then 𝐵 and 𝐶 are disjoint.
• Principle of inclusion-exclusion
• 𝐵 ∪ 𝐶 = 𝐵 + 𝐶 − 𝐵 ∩ 𝐶

Set Operations

• Set Difference, 𝐵 − 𝐶
• 𝐵 − 𝐶 = 𝑦 ∣ 𝑦 ∈ 𝐵 ∧ 𝑦 ∉ 𝐶
• 8,6,7,5,3,0,9 − 0,2,4,6,8 = ?
• Complement, 𝐵
• 𝐵 = 𝑦 ∣ 𝑦 ∉ 𝐵
• 𝐵 = 𝑉 − 𝐵
• How can we express set difference using the other set operations?
• 𝐵 − 𝐶 = 𝐵 ∩ 𝐶

Venn Diagrams

• Draw a rectangle to

represent 𝑉.

• Draw geometric shapes

inside to represent sets.

• Use points to represent

particular elements.

𝑉 𝑊

a e i

• u

Venn Diagrams

= 𝐵 𝐵 𝑉

Venn Diagrams

= 𝐵 ∪ 𝐶 𝐵 𝑉 𝐶

Venn Diagrams

= 𝐵 ∩ 𝐶 𝐵 𝑉 𝐶

Venn Diagrams

= 𝐵 − 𝐶 𝐵 𝑉 𝐶

Set Identities

• Same as for propositional logic!
• Identity, domination, idempotent, double negation, commutativity,

associativity, distributivity, DeMorgan, absorption, negation

• Operators map from logic to sets
• 𝑞 ∨ 𝑟 ⇒ 𝑄 ∪ 𝑅
• 𝑞 ∧ 𝑟 ⇒ 𝑄 ∩ 𝑅
• ¬𝑞 ⇒ 𝑄
• 𝑞 → 𝑟 ⇒ 𝑄 ⊆ 𝑅
• 𝑞 ↔ 𝑟 ⇒ 𝑄 = 𝑅

Set Identities

• Ex: 𝐵 ∩ 𝐶 = 𝐵 ∪ 𝐶

(DeMorgan)

• Show 𝐵 ∩ 𝐶 ⊆ 𝐵 ∪ 𝐶
• Show 𝑦 ∈ 𝐵 ∩ 𝐶 → 𝑦 ∈ 𝐵 ∪ 𝐶
• Assume 𝑦 ∈ 𝐵 ∩ 𝐶
• Then, 𝑦 ∉ 𝐵 ∩ 𝐶
• ¬ 𝑦 ∈ 𝐵 ∩ 𝐶
• ¬ 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶
• ¬ 𝑦 ∈ 𝐵 ∨ ¬ 𝑦 ∈ 𝐶
• 𝑦 ∉ 𝐵 ∨ 𝑦 ∉ 𝐶
• 𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶
• 𝑦 ∈ 𝐵 ∪ 𝐶
• □
• Show 𝐵 ∪ 𝐶

⊆ 𝐵 ∩ 𝐶

• Show 𝑦 ∈ 𝐵 ∪ 𝐶

→ 𝑦 ∈ 𝐵 ∩ 𝐶

• Assume 𝑦 ∈ 𝐵 ∪ 𝐶
• Then, 𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶
• 𝑦 ∉ 𝐵 ∨ 𝑦 ∉ 𝐶
• ¬ 𝑦 ∈ 𝐵 ∨ ¬ 𝑦 ∈ 𝐶
• ¬ 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶
• ¬ 𝑦 ∈ 𝐵 ∩ 𝐶
• 𝑦 ∉ 𝐵 ∩ 𝐶
• 𝑦 ∈ 𝐵 ∩ 𝐶
• □

Truth tables ⇒ Membership tables

𝑩 𝑪 𝑩 ∩ 𝑪 𝑩 ∩ 𝑪 𝑩 𝑪 𝑩 ∪ 𝑪 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Verify that 𝐵 ∩ 𝐶 = 𝐵 ∪ 𝐶

XOR

• 𝐵 ⊕ 𝐶: elements in 𝐵, or in 𝐶, but not in both.
• Symmetric difference
• Draw the Venn diagram
• Claim: 𝐵 ⊕ 𝐶 = 𝐵 ∪ 𝐶 − 𝐵 ∩ 𝐶
• Proof?