Sets

Definition

A set is a collection of anything: numbers, letters, objects, etc.

The members, or elements, of the sets are written between braces like this: {1, 2, 3, 4, 5}.

The elements of this set are simply the numbers 1, 2, 3, 4, and 5. Another example of a set is {apples, peaches, pears}.

Two sets are equal

Two sets are equal if they have the same elements. The order in which the elements of the set are listed does not matter. Thus {1, 2, 3, 4, 5} = {5, 4, 3, 2, 1}. We can use one letter to stand for a whole set; for example, A = {1, 2, 3, 4, 5}.

The union of two sets

– The union of two sets is a set, which contains all elements of two sets.

– The union of sets A and B is written A∪B.

Example:

If A = {1, 2, 3, 4} and B = {2, 4, 6}, find A∪B.

All the elements in either A or B or both are 1, 2, 3, 4, and 6.

Therefore A∪B = {1, 2, 3, 4, 6}.

The intersection of two sets

– The intersection of two sets is a set, which contains the elements in both two sets

– The intersection of sets A and B is written A∩B.

Example:

If A = {1, 2, 3, 4} and B = {2, 4, 6}, find A∩B.

The elements in both A and B are 2 and 4.

Therefore AB = {2, 4}.

If two sets have no elements in common, then their intersection is the null or empty set written as ∅ or { }.

Example:

If A= {1, 3, 5, 7} and B = {2, 4, 6, 8}, find AB.

There is no elements in common in both A and B.

 Therefore AB = {}.

To perform several union and intersection operations

First operate on sets within parentheses.

Example:

If A = {1, 2, 3} and B = {2, 3, 4, 5, 6} and C = {1, 4, 6} find A(BC).

First we find BC by listing all the elements in both B and C. BC = {4, 6}.

Then A(BC) is just the set of all members in at least one of the sets A and {4, 6}.

Therefore, A(BC) = {1, 2, 3, 4, 6}.

A subset

– A subset of a set is a set, all of whose members are in the original set. Thus, {1, 2, 3} is a subset of the set {1, 2, 3, 4, 5}.

– Note that the null set is a subset of every set, and also that every set is a subset of itself.

In general, a set with n elements has 2n subsets.

Example:
How many subsets does {x, y, z} have?

This set has 3 elements and therefore 23, or 8 subsets.

They are: {x}, {y}, {z}, {x,y}, {y,z}, {x,z}, {x,y,z}, {∅}