Discrete Mathematics Discrete Mathematics -- Chapter 3: Set Theory - - PowerPoint PPT Presentation

Discrete Mathematics Discrete Mathematics

• - Chapter 3: Set Theory

Hung-Yu Kao (高宏宇) Department of Computer Science and Information Engineering, N l Ch K U National Cheng Kung University

Outline

3.1 Set and Subsets 3 2 Set Operations and the Laws of Set Theory 3.2 Set Operations and the Laws of Set Theory 3.3 Counting and Venn Diagrams 3.4 A First Word on Probability 3 5 The Axioms of Probability 3.5 The Axioms of Probability 3.6 Conditional Probability: Independence 3.7 Discrete Random Variables

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Why Set?

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3.1 Set and Subsets

Set: should be a well-defined collection of objects. Elements (members): These objects are called elements or

members of the set members of the set.

• could be another set, 1 ≠ {1} ≠ {{1}}

Capital letters represent sets: A, B, C

p p lowercase letters represent elements: x, y

E.g.,

A et be de i ted b li ti it ele e t ithi et b e

B y A x ∉ ∈ ,

A set can be designated by listing its elements within set braces

“{“,”}”.

E.g., A = {1, 2, 3, 4, 5}, B = {x | x is an integer, and 1 ≤ x ≤ 5}

g { } { | g }

Cardinality (size): |A| denotes the number of elements in A. for finite sets

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Set and Subsets

Universe (Universe of discourse): h denotes the range of all

elements to form any set. fi i i 3 1 f C d f i

Definition 3.1: If C and D are sets from a universe h

• Subset:

, if every element of C is an element of D.

• Proper subset:

if in addition D contains an

) ( C D D C ⊇ ⊆ ) ( C D D C ⊂ ⊂

• Proper subset: , if, in addition, D contains an

element that is not in C.

• )

( C D D C ⊂ ⊂ ] [ D x C x x D C ∈ ⇒ ∈ ∀ ⇔ ⊆

] [ )

• f

subset a not is i.e., ( D x C x x D C D C ∈ ⇒ ∈ ∀ ⇔ ⊄ ] ) ( [ ] [ ] [ D x C x x D x C x x D x C x x ∈ ∨ ∈ ¬ ¬ ∃ ⇔ ∈ ⇒ ∈ ¬ ∃ ⇔ ∈ ⇒ ∈ ¬∀ ⇔ ] [ )] ( ) ( [ ] ) ( [ D x C x x D x C x x x C x x ∉ ∧ ∈ ∃ ⇔ ∈ ¬ ∧ ∈ ¬¬ ∃ ⇔ ⇔

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Set and Subsets

Definition 3.2: The sets C and D are equal for a given universe h ,

) ( ) ( C D D C D C ⊆ ∧ ⊆ ⇔ =

Let A, B, C h ,

C A C B B A C A C B B A ⊂ ⊆ ⊂ ⊆ ⊆ ⊆ h d f ) then , and If ) b then , and If ) a

⊆

C A C B B A C A C B B A ⊂ ⊂ ⊂ ⊂ ⊂ ⊆ then , and If ) d then , and If ) c

Let h = {1, 2, 3, 4, 5} with A = {1, 2, 3}, B = {3, 4}, and

C = {1, 2, 3, 4}. Then the following subset relations hold:

• A

A A B A A C B C A C A ⊄ ⊄ ⊆ ⊂ ⊂ ⊆ ) f ) e ) d ) c ) b ) a

A is not a proper subset of A

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Set and Subsets

Null set (empty set), ∅ or { }: is the set containing no elements.

• |∅|=0 but {0} ≠∅
• ∅ ≠{∅}
• ∅ ≠{∅}

Power set, P(A): is the collection (set) of all subsets of the set A

from universe h .

Example: A = {1, 2, 3}

• P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, A}

F fi it t A ith |A|

For any finite set A with |A|=n A has 2n subsets and |P(A)|= 2n There are

subsets of size k 0≤k≤n

⎞ ⎜ ⎜ ⎛n

There are subsets of size k, 0≤k≤n Counting the subsets of A (binomial theorem)

⎠ ⎜ ⎜ ⎝k

n n

n n n n 2 ∑ ⎞ ⎜ ⎜ ⎛ ⎞ ⎜ ⎜ ⎛ + + ⎞ ⎜ ⎜ ⎛ + ⎞ ⎜ ⎜ ⎛

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n n k

k n 2 1 = ∑ ⎠ ⎜ ⎜ ⎝ = ⎠ ⎜ ⎜ ⎝ + ⋅ ⋅ ⋅ + ⎠ ⎜ ⎜ ⎝ + ⎠ ⎜ ⎜ ⎝

=

Set and Subsets

Theorem 3.2

? } { ? } { φ φ φ φ ? } { ? } { φ φ φ φ ⊂ ⊆ ? ? φ φ φ φ ⊂ ⊆

T T T F

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Set and Subsets

• Ex 3.9:

Gray code: There is exactly one bit that changes from one

bi i h

Another Gray code

binary string to the next one.

0 Φ 1 { } 00 0 Φ 10 0 { } 000 100 000 010 000 001

Another Gray code

1 {x} (a) 10 0 {x} 11 0 {x, y} 01 0 {y} 100 110 010 010 011 001 001 101 100 0 0 Φ 01 0 {y} 01 1 {y, z} 11 1 {x, y, z} 010 011 111 001 101 111 100 110 010 0 0 Φ 1 0 {x} 1 1 {x, y} 10 1 {x, z} 00 1 {z} (c) 101 001 (d) 110 100 (e) 011 111 (f) { y} 0 1 {y} (b)

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Set and Subsets

Ex 3.12

{ }

subsets 1 are There elements. r contain A that

• f

subsets all consider and , , , , A

2 1

⎞ ⎜ ⎜ ⎛ + ⎞ ⎜ ⎜ ⎛ = ⎞ ⎜ ⎜ ⎛ + = n n n a a a x Let

n

L subsets. 1 are There ⎠ ⎜ ⎜ ⎝ − + ⎠ ⎜ ⎜ ⎝ = ⎠ ⎜ ⎜ ⎝ r r r

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Set and Subsets

E 3 13

Ex 3.13

10 inequality the

• f

solutions integer e nonnegativ

• f

number the

6 2 1

< + + + x x x L

⎞ ⎛ + ⎞ ⎛ + = + + + ≤ ≤ ∀ k k k x x x k k 5 1 6 is

• solution t
• f

number the , 9 ,

6 2 1

L ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + k k k k 5 1 6 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + 9 15 9 1 9 7 1, chapter in

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Set and Subsets

, N N +

+ + ⊆ Q

Z Q

+ +

= ∩ R C R Q R ⊆

+

Z Z Q = ∩

* + + +

= ∪ R R Z

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3.2 Set Operations and the Laws of Set p Theory

Definition 3.5: For A and B ⊆ h

} | { ) and

• f

union (the ) a B x A x x B A B A ∈ ∨ ∈ = ∪ ) and

• f

difference symmetric (the ) c } | { ) and

• f
• n

intersecti (the ) b B A B A B x A x x B A B A ∆ ∈ ∧ ∈ = ∩ } | { } ) ( | { B A x B A x x B A x B x A x x ∩ ∉ ∧ ∪ ∈ = ∩ ∉ ∧ ∈ ∨ ∈ =

Definition 3.6: The sets S, T ⊆ h , are called disjoint

(mutually disjoint) when

φ = ∩T S

(mutually disjoint), when

. φ = ∩T S

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Set Operations and the Laws of Set p Theory

Theorem 3.3: If S, T ⊆ h are disjoint if and only if Proof:

T S T S ∆ = ∪

) 1 ( T S T S ∪ ⇒ ∪

Proof:

, i.e., disjoint, and But ) 1 ( T S T S T S x T S x T S T x S x T S x ∆ ∆ ∈ ⇒ ∩ ∉ ∈ ∪ ∈ ⇒ ∪ ∈ ) 2 ( T y S y T S y T S T S ∈ ∪ ∈ ⇒ ∆ ∈ ∆ ⊆ ∪ ∴ T S T S T S y ∪ ⊆ ∆ ∴ ∪ ∈ ⇒ and T S T S T S T S T S T S ∆ = ∪ ∴ ∪ ⊆ ∆ ∆ ⊆ ∪ Q

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Prove the converse by the method of proof by contradiction

Set Operations and the Laws of Set p Theory

Definition 3.7: For a set A ⊆ h , the complement of A, denoted

h – A or , is given by h

} | { A x x x ∉ ∧ ∈

A

Definition 3.8: For A, B ⊆ h , the (relative) complement of A in B,

denoted B – A or, is given by

Ex 3 18 : For h = R A = [1 2] B = [1 3)

} | { A x B x x ∉ ∧ ∈

Ex 3.18 : For h = R, A = [1, 2], B = [1, 3)

• B

A B x x x x A ∪ = < ≤ ⊆ ≤ ≤ = ? ) b } 3 1 | { } 2 1 | { ) a

} 3 1 | { B x x < ≤

A B B A B A = +∞ ∪ −∞ ⊆ +∞ ∪ −∞ = = ∩ = ∪ ) , 2 ( ) 1 , ( ) , 3 [ ) 1 , ( ) d ? c) ? ) b

} 3 1 | { B x x = < ≤ = } 2 1 | { A x x = ≤ ≤ =

Theorem 3.4: The following statements are equivalent:

A B +∞ ∪ ∞ ⊆ +∞ ∪ ∞ ) , 2 ( ) 1 , ( ) , 3 [ ) 1 , ( ) d

(d) (c) ) (b ) (a A B A B A B B A B A ⊆ = ∩ = ∪ ⊆

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(d) (c) ) (b ) (a A B A B A B B A B A ⊆ ∩ ∪ ⊆

Sets and Logic

L i d h ll h Logic and set theory go very well together. The previous definitions can be made very succinct: A if d l if ( A) x∉A if and only if ¬(x∈A) A⊆B if and only if (x∈A → x∈B) is True (A∩B) if d l if ( A B) x ∈ (A∩B) if and only if (x∈A ∧ x∈B) x ∈ (A∪B) if and only if (x∈A ∨ x∈B) x ∈ A B if and only if (x∈A ∧ x∉B) x ∈ A–B if and only if (x∈A ∧ x∉B) x ∈ A ∆ B if and only if (x∈A ∧ x∉B) ∨ (x∈B ∧ x∉A) x ∈ A if and only if ¬(x∈A) x ∈ A if and only if ¬(x∈A) X ∈P(A) if and only if X⊆A

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The Laws of Set Theory

= A A plement Double Com :

• f

Law ) 1 ⎩ ⎨ ⎧ ∩ = ∩ ∪ = ∪ ⎩ ⎨ ⎧ ∪ = ∩ ∩ = ∪ A B B A A B B A e Commutativ B A B A B A B A DeMogran's plement

• uble Com

: Laws ) 3 : Laws ) 2 :

• w

) ⎧ ∪ ∩ ∪ = ∩ ∪ ⎩ ⎨ ⎧ ∩ ∩ = ∩ ∩ ∪ ∪ = ∪ ∪ C A B A C B A C B A C B A C B A C B A e Associativ ) ( ) ( ) ( ) ( ) ( ) ( ) ( : Laws ) 4 ⎨ ⎧ = ∪ ⎨ ⎧ = ∪ ⎩ ⎨ ⎧ ∩ ∪ ∩ = ∪ ∩ ∪ ∩ ∪ = ∩ ∪ A A Identity A A A Idempotent C A B A C B A C A B A C B A ve Distributi : Laws 7) : Laws ) 6 ) ( ) ( ) ( ) ( ) ( ) ( : Laws ) 5 φ ⎩ ⎨ ⎧ = ∩ = ∪ ⎩ ⎨ ⎧ = ∩ = ∪ ⎩ ⎨ = ∩ ⎩ ⎨ = ∩ A U U A A A U A A Inverse A U A Identity A A A Idempotent : Laws Domination ) 9 : Laws 8) : Laws 7) : Laws ) 6 φ φ φ ⎩ ⎨ ⎧ = ∪ ∩ = ∩ ∪ ⎩ = ∩ ⎩ = ∩ A B A A A B A A Absorption A A A ) ( ) ( : Laws ) 10 φ φ φ

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The Laws of Set Theory

Ex 3.20

Simplify the expression

B C B A ∪ ∩ ∪ ) (

p y p

B C B A ∪ ∩ ∪ ) ( ? and

• f

in terms B

• A

expressing about How ∪

B A∩ B A∪ =

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p g

Set Operations and the Laws of Set p Theory

Definition 3.9: The dual of s, sd can be replaced

mutually.

• h

Theorem 3.5: The Principle of Duality, let s denote

th d li ith th lit f t t

and (2) and ) 1 ( φ ∩ ∪

a theorem dealing with the equality of two set

• expressions. Then sd is also a theorem.

Ex 3.19 : find a dual for statement A⊆ B (Th. 3.4)

A∪B=B A∩B=B (duality) A∪B B A∩B B (duality)

But A∩B=B ⇔ B⊆A (the dual of A⊆ B)

A∩B=A A∪B=A ⇔ B⊆A

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Set Operations and the Laws of Set Theory

Theorem 3.6: Generalized DeMorgan’s Laws,

let I be an index set, then let I be an index set, then

P f

U I I U

I i i I i i I i i I i i

A A A A

∈ ∈ ∈ ∈

= = (2) ) 1 (

Proof:

I U U

i i i i i

A x I i A x I i A x A x A x ∈ ⇔ ∈ ∈ ⇔ ∈ ∉ ⇔ ∉ ⇔ ∈ all for all for ) 1 (

I i i i i I i i I i i ∈ ∈ ∈

) (

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The Use of Sets

Sets are the most simple yet non-trivial structures in Sets are the most simple, yet non-trivial structures in

mathematics.

Many other mathematical objects and properties can be Many other mathematical objects and properties can be

defined by them.

For us, sets are useful to understand the principles of

ti d b bilit th counting and probability theory.

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3.3 Counting and Venn Diagrams

Venn diagrams are used to depict the various Venn diagrams are used to depict the various

unions, subsets, complements, intersections etc. f t

• f sets:

A∩B A B

John Venn

B C

John Venn

(1834~1923)

C

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Venn Diagrams

B A∩ A B B A∪

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Remember the Sum Rule

If we have sets of mutually disjoint events If we have sets of mutually disjoint events E1, E2, …, Em, where Ej can occur in nj ways (with Ej∩Ek = ∅ for all j≠k),

j

then there are n1+n2+…+nm possible events. Think union of sets: Let |Ej| = nj for all 1≤j≤m, th E E E h + + + l t then E1∪E2, …∪Em has n1+n2+…+nm elements. E1 E2 E3

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Inclusion and Exclusion (排容原理)

The Principle of Inclusion and Exclusion

generalizes the Sum Rule to the cases where the generalizes the Sum Rule to the cases where the events are not disjoint.

We can use it when solving counting problems…

g g p

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3.3 Counting and Venn Diagrams

| | | | | | | | ) 1 ( B A B A B A | | | | | | | | (2) | | | | | | | | ) 1 ( B A U B A B A B A B A B A ∪ − = ∪ = ∩ ∩ − + = ∪ | | | | (4) | | | | | | | | | | | | | | | | ) 3 ( | | | | | | | | C B A C B A C B A C B C A B A C B A C B A B A B A U ∪ ∪ = ∩ ∩ ∩ ∩ + ∩ − ∩ − ∩ − + + = ∪ ∪ ∩ + − − = | | | | | | | | | | | | | | | | | | | | | | | | (4) C B A C B C A B A C B A U C B A U C B A C B A ∩ ∩ − ∩ + ∩ + ∩ + − − − = ∪ ∪ − = ∪ ∪ ∩ ∩

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How to Derive the 3 Set Case

Using the various Laws for Sets: |A∪B∪C| = |A∪(B∪C)| |A∪B∪C| |A∪(B∪C)| = |A| + |B∪C| – |A∩(B∪C)| |A| + |B| + |C| |B∩C| |A∩(B∪C)| = |A| + |B| + |C| – |B∩C| – |A∩(B∪C)| = |A| + |B| + |C| – |B∩C| – |(A∩B)∪(A∩C)| = |A| + |B| + |C| – |B∩C| – |A∩B| – |A∩C| + |A∩B∩A∩C| = |A| + |B| + |C| – |B∩C| – |A∩B| – |A∩C| + |A∩B∩C|

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| |

Examples

1到250的整數中 ，可被3或 5或 7 整除的數有

幾個? 幾個?

Ans:

假設能分別被3 5 7 整除的數字集合分別為 A B C

假設能分別被3, 5, 7,整除的數字集合分別為 A, B, C

則根據排容原理, |A∪B∪C |= |A| + |B| + |C| - |A∩B| |A∩C| |B∩C| + |A∩B∩C| = 83+50+35 16

• |A∩C| - |B∩C| + |A∩B∩C| = 83+50+35 – 16 –

11 – 7 + 2 = 136

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Examples

Throw two dice, how many ways of throwing: Throw two dice, how many ways of throwing: a total of 8, value 6+not value 6, or two identical values? Counting the possibilities: a total of eight: 5 value 6+not value 6: 10 two identical values: 6

Total: 5+10+6–2–1–0+0 = 18

two identical values: 6 total 8 and 6+not 6: 2 total 8 and two id-vs: 1 d id

“total 8”

6+not 6 and two id-vs: 6+not 6, two id-vs, total 8: 0

2 1 “id vs” 8 5 2 id-vs

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“6+not 6”

3.4 Probability

Under the assumption of equal likelihood, let Φ be a

sample space for an experiment Ε. Any subset A of Φ is ll d t E h l t f Φ i ll d called an event. Each element of Φ is called an elementary event, so if |Φ| = n and a ∈ Φ, A ⊆ Φ, then

Pr(a) = The probability that {a} occurs Pr(a) = The probability that {a} occurs

= , and

{ }

1 a Φ =

{ }

a 1 =

Pr(A) = The probability that A occurs =

n Φ A A Φ =

n Φ A

We often write Pr(a) for Pr({a})

n Φ

n

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Probability

• Ex 3.29 :
• If Dillon rolls a fair die, what is the probability he gets (a) a 5 or a 6, (b)

an even number? an even number?

Sample space Φ={1, 2, 3, 4, 5, 6}.

(a) event A={5, 6} Pr(A)=|A|/|Φ|= 2/6=1/3. (b) event B ={2, 4, 6} Pr(B)=|B|/|Φ|=3/6=1/2. Furthermore we also notice here that Furthermore we also notice here that i) Pr(Φ) =| Φ |/|Φ|= 6/6 =1—after all, the occurrence of the event is a certainty; and ii) Pr(A) =Pr({1, 2, 3, 4}) =|A|/Φ = 4/6=2/3=1 − 1/3 = 1 − Pr(A).

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Probability

Ex 3.37 :

In a survey of 120 passengers, an airline found that 48 enjoyed

wine with their meals, 78 enjoyed mixed drinks, and 66 enjoyed iced tea. In addition, 36 enjoyed any given pair of these beverages and 24 passengers enjoyed them all If two passengers beverages and 24 passengers enjoyed them all. If two passengers are selected at random from the survey sample of 120, what is the probability that

a) (Event A) they both want only iced tea with their meals?

) ( ) y y

b) (Event B) they both enjoy exactly two of the three

beverage offerings?

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g g

Probability

The sample space Φ consists of the pairs of passengers we

can select from the sample of 120, so

The Venn diagram indicates that there are 18 passengers

who drink only iced tea, |A| d P (A) 51/ 2380

C(120,2)

18

so |A|= and Pr(A) =51/ 2380. |B|= Pr(B) =3/ 34.

18 2

C

36 2

C

W:48, M:78, T:66 W∩T:36, W∩M:36, T∩M:36 W∩T∩M:24

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3.5 Probability Rules and Laws

− = A) Pr( 1 ) A ( Pr : Complement

• f

Rule The ) 1

A A

∩ + = ∪ ∩ − + = ∪ B A B A B A B A B A B A B A ) Pr( ) Pr( ) Pr( ) ( Pr disjoint, are , When ) Pr( ) Pr( ) Pr( ) ( Pr : Rule Additive The ) 2

A B A

= = ∩ ∩ = B A B A B A B A B B A B A ) | Pr( ) Pr( ) | Pr( ) Pr( ) ( Pr : Rule tive Multiplica The ) 4 ) Pr( ) Pr( ) | Pr( : y Probabilit l Conditiona ) 3 + = = ∩ A B A A B A B B A B A B A ) | Pr( ) Pr( ) | Pr( ) Pr( ) Pr( : y Probabilit Total

• f

Law ) 5 ) Pr( ) Pr( ) ( Pr t, independen are , When φ + = ∩ = = ∩ ∑ =

= j i n i i i

A B A A B A A B A B B A B A A A A B A B ) | Pr( ) Pr( ) | Pr( ) Pr( ) | Pr( ) Pr( ) Pr( ) Pr( ) | Pr( : Theorem Bayes' ) 6 , ) | Pr( ) Pr( ) Pr( : version) (Extended

1

φ = ∩ ∑ = ∩ = +

= j i n i i i k k k k

A A A B A A B A B B A B A A B A A B A B , ) | Pr( ) Pr( ) | Pr( ) Pr( ) Pr( ) Pr( ) | Pr( : version) (Extended ) | Pr( ) Pr( ) | Pr( ) Pr( ) Pr(

1

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i 1

Exercises

3.1: 2, 20 3 2: 4 8 3.2: 4, 8 3.3: 6

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