6.1 Set Theory - Definitions and the Element Method of Proof

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Discrete Structures

Chapter 6: Set Theory

6.1 Set Theory: Definitions and the Element Method of Proof

The introduction of suitable abstractions is our only mental aid to organize and master complexity.– E. W. Dijkstra, 1930 – 2002

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6.1 Set Theory - Definitions and the Element Method of Proof

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Subsets

• Let’s write what it means for a set A to be a subset of a set B as a formal universal conditional statement:

A B x, if x A then x B.

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6.1 Set Theory - Definitions and the Element Method of Proof

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Subsets

• The negation is existential

A B x, if x A and x B.

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6.1 Set Theory - Definitions and the Element Method of Proof

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Subsets

• A proper subset of a set is a subset that is not equal to its containing set.

A is a proper subset of B 1. A B, and 2. there is at least one element

in B that is not in A.

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6.1 Set Theory - Definitions and the Element Method of Proof

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Element Argument

• Let sets X and Y be given. To prove that X Y,

1. Suppose that x is a particular but arbitrarily chosen element of X,

2. Show that x is an element of Y.

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6.1 Set Theory - Definitions and the Element Method of Proof

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Example – pg. 350 # 4

• Let A = {n | n = 5r for some integer r} and B = {m | m = 20s for some integer s}.

a. Is A B? Explain.

b. Is B A? Explain.

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Set Equality

• Given sets A and B, A equals B, written A = B, iff every element of A is in B and every element of B is in A. Symbolically,

A = B A B and B A

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6.1 Set Theory - Definitions and the Element Method of Proof

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Operations on Sets

• Let A and B be subsets of a universal set U.

1. The union of A and B denoted A B, is the set of all elements that are in at least one of A or B.

Symbolically: A B = {x U | x A or x B}

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Operations on Sets

• Let A and B be subsets of a universal set U.

2. The intersection of A and B denoted A B, is the set of all elements that are common to both A or B.

Symbolically: A B = {x U | x A and x B}

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6.1 Set Theory - Definitions and the Element Method of Proof

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Operations on Sets

• Let A and B be subsets of a universal set U.

3. The difference of B minus A (or relative complement of A in B) denoted B – A, is the set of all elements that are in B but not A.

Symbolically: B – A = {x U | x B and x A}

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6.1 Set Theory - Definitions and the Element Method of Proof

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Operations on Sets

• Let A and B be subsets of a universal set U.

4. The complement of A denoted Ac, is the set of all elements in U that are not A.

Symbolically: Ac = {x U | x A}

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Example – pg. 350 # 11

• Let the universal set be the set R of all real numbers and let

A = {x R | 0 < x 2}, B = {x R | 1 x < 4}, and C = {x R | 3 x < 9}. Find each of the following:

a. A B b. A B c. Ac

d. A C e. A C f. Bc

g. Ac Bc h. Ac Bc i. (A B)c

j. (A B)c

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Unions and Intersections of an Indexed Collection of Sets

Given sets A0, A1, A2, … that are subsets of a universal set U and given a nonnegative integer n,

0

0

0

0

| for at least one 0,1,2,...,

| for at least one nonnegative integer

| for all 0,1,2,...,

| for all nonnegative integers

n

i ii

i ii

n

i ii

i ii

A x U x A i n

A x U x A i

A x U x A i n

A x U x A i

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Definitions

• Empty Set

A set with no elements is called the empty set (or null set) and denoted by the symbol .

• Disjoint

Two sets are called disjoint iff they have no elements in common. Symbolically:

A and B are disjoint A B =

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Definitions

• Mutually Disjoint

Sets A1, A2, A3, … are mutually disjoint (or pairwise disjoint or nonoverlapping) iff no two sets Ai and Aj with distinct subscripts have any elements in common. More precisely, for all i, j = 1, 2, 3, …

Ai Aj = whenever i j.

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Example – pg. 305 # 23

Let for all

positive integers i.

1 1 1 1| ,iV x x

i i i i

4 4

1 1

1 2 3

1 1

1 1

a. ? b. ?

c. Are , , ,... mutually disjoint? Explain.

d. ? e. ?

f. ? g. ?

i ii i

n n

i ii i

i ii i

V V

V V V

V V

V V

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Definition

• Partition

A finite or infinite collection of nonempty sets {A1, A2, A3, …} is a partition of a set A iff,

1. A is the union of all the Ai

2. The sets A1, A2, A3, …are mutually disjoint.

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Example – pg. 351 # 27

b. Is , , , , , , , a partition of

, , , , , , , ?

c. Is 5, 4 , 7, 2 , 1,3, 4 , 6,8 a partition of

1, 2,3, 4,5,6,7,8 ?

e. Is 1,5 , 4,7 , 2,8,6,3 a partition of

1, 2,3, 4,5,6,7,8 ?

w x v u y q p z

p q u v w x y z

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Definition

• Power Set

Given a set A, the power set of A is denoted (A), is the set of all subsets of A.

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Example – pg. 351 # 31

• Suppose A = {1, 2} and B = {2, 3}. Find each of the following:

a. b.

c. d.

A B A

A B A B

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