Chapter 2

Venn diagrams

A Venn diagram is a picture of the relationships among certain sets.

Sets

A set is a collection of objects. The objects are called elements. Examples: {1, 3, 5, 7, 9} {a, e, i, o, u} {red, blue, yellow}

Number of proper subsets

A set with n elements has 2^n - 1 proper subsets.

The number of subsets

A set with n elements has 2n subsets

Notation

Capital letters represent sets Curly brackets enclose the elements of a set Lowercase letters represent set elements A long list can be abbreviated using ..., which is also called an ellipsis. For example, the set A = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18} can be shortened to A = {0, 2, 4, ..., 16, 18} The empty set is a set with no elements. The notation for the empty set is Ø or { } but not {Ø}.

Element notation

E = {a, b, c, d, ..., x, y, z} We use the notation g E to mean that g is an element of the set E. On the other hand, we use the notation E to mean that is not an element of E.

Cardinal number of a set

The cardinal number of a set is the number of elements belonging to the set. The symbol n(A) refers to the cardinal number of the set A. Cardinal numbers can be finite or infinite Examples: K = {6, 8, 10} B = {3, 4, 5, ..., 20} M = {0} Ø n(K) = 3 n(B) = 18 n(M) = 1 n(Ø) = 0

The complement of a set

The collection of elements not belonging to the set A is called the complement of A. This complement of A is denoted A'.

The difference of two sets Venn

The difference of sets A and B is the set of all elements which belong to set A but do not belong to set B

Intersection of sets Venn

The intersection of sets A and B is the set of elements which belong to both sets.

Subsets

The set A is a subset of the set B if every element of A is an element of B.

Union of sets Venn

The union of sets A and B is the set of elements which belong to either set (or both).

Disjoint sets

Two sets A and B are called disjoint sets if their intersection is empty

Set equality

Two sets A and B are equal if the following two conditions are met: every element of A is an element of B every element of B is an element of A.

Proper subsets

We say that A is a proper subset of B if 1) A is a subset of B and 2) A is not equal to B Examples: {3, 4, 5, 6} {1, 2, 3, 4, 5, 6, 8, 9} {1, 2, 3, 4} {2, 3, 5, 7, 9} {3, 4, 5, 6, 8} {3, 4, 5, 6}

Cartesian products

When we have two sets A and B, we can create a new set called the Cartesian product A x B (read "A cross B") Example: Given A = {1, 2} and B = {3, 4} A x B = {(1, 3), (1, 4), (2, 3), (2, 4)} B x A = {(3, 1), (3, 2), (4, 1), (4, 2)} Note that A x B ≠ B x A

Word description

the set of odd counting numbers less than ten

Integers

{..., -3, -2, -1, 0, 1, 2, 3, ...}

Whole numbers

{0, 1, 2, 3, 4, ...}

Natural or counting numbers

{1, 2, 3, 4, ...}

Listing Method

{1, 3, 5, 7, 9}

How many subsets does {a, b, c} have?

{a, b, c} {a} {a, b} {b} {a, c} {c} {b, c} Ø It has 8 subsets.

Rationals

{x | x = p/q where p & q are integers and q ≠ 0}

Set builder-notation

{x | x is an odd counting number less than 10}