Section 3.1

Sets and their operation

Definitions

A set S is collection of objects. These objects are said to be members or elements of the set, and the shorthand for writing “x is an element of S” is “x S.”

The easiest way to describe a set is by simply listing its elements (the “roster method”). For example, the collection of odd one-digit numbers could be written {1, 3, 5, 7, 9}. Note that this is the same as the set {9, 7, 5, 3, 1} since the order elements are listed does not matter in a set.

Examples

The elements of a set do not have to be numbers as the following examples show:

1. {Doug, Amy, John, Jessica}

2. {TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF}

3. { {A,B}, {A,C}, {B,C} }

4. { }

Common sets of numbers

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N … set of natural numbers {0, 1, 2, …}Z … set of integers {…, -2, -1, 0, 1, 2, …}Q … set of rational numbersR … set of real numbers

DefinitionsIf A and B are sets, then the notation A B (read

“A is a subset of B”) means that every element of set A is also an element of set B.

Practice. Which is true?1. {1, 2, 3, 4} {2, 3, 4}2. Z Q3. Z N4. { } {a, b, c}5. {3, 5, 7} {2, 3, 5, 7, 11}6. {a, b} { {a, b}, {a, c}, {b, c} }7. {a} { {a, b}, {a, c}, {a, b, c} }

Set notation

Large sets cannot be listed in this way so we need the more compact “set-builder” notation. This comes in two types exemplified by the following:

1. (Property) {n Z : n is divisible by 4}

2. (Form) {4k : k Z}

Practice with property description

List five members of each of the following sets:

1. {n N : n is an even perfect square }

2. {x Z : x – 1 is divisible by 3 }

3. {r Q : r2 < 2 }

4. {x R : sin(x) = 0 }

Practice with form description

List five members of each of the following sets:

1. { 3n2 : n Z }

2. { 4k + 1 : k N }

3. { 3 – 2r : r Q and 0 r 5 }

Definitions of set operationsLet A and B be sets with elements from a specified

universal set U.

A B (read “A intersect B”) is the set of elements in both sets A and B.

A B (read “A union B”) is the set of elements in either set A or B.

A – B (read “A minus B”) is the set of elements in set A which are not in B.

A’ (read “the complement of A”) is the set of elements in the universe U which are not in A.

Practice with set operations

Let A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10}, C = {2, 3, 5, 7}, D = {6, 7, 8, 9, 10}

be sets with elements from the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Find each of the following:

1. A C

2. B D

3. B – D

4. B’

5. (A B) – C

6. (A C) B

7. B’ C’

8. (B C)’

9. (C D) – A

10.B D’

Venn diagrams

Inclusion-Exclusion Principle

The notation n(A) means “the number of elements of A.” For example, if A = {2, 3, 6, 8, 9}, then n(A) = 5.

Principle of Inclusion/Exclusion for two sets A and B:

n(A B) = n(A) + n(B) – n(A B)

Inclusion-Exclusion Principle

Example. A = { 2, 4, 6, 8, …, 96, 98, 100 } and B = { 5, 10, 15, 20, …, 90, 95, 100}

n(A B) = n(A) + n(B) – n(A B)

= 50 + 20 – 10

= 60

Inclusion-Exclusion Principle

Principle of Inclusion/Exclusion for three sets A, B, and C:

n(A B C) = n(A) + n(B) + n(C)

– n(A B) – n(A C) – n(B C)

+ n(A B C)