# Operation of sets I

__Union____ of Sets__

From two given sets A and B we can make a new set that consists of all the elements of A and all the elements of B. This new set is called the union of A and B. It is represented by the symbol A + B.

For example, let A = {a, b, c, d} and let B = {c, d, e}. Then A + B = {a, b, c, d, e}. Notice that the elements c and d are in A as well as B, but they are written only once in the list for A Ãˆ B.

The union of two sets is defined in symbols as follows:

A Ãˆ B = {x: x is in A or x is in B}.

__Intersection of Sets__

From two given sets A and B we can make a new set that consists of all the elements that belong to both A and B at the same time. This new set is called the intersection of A and B. It is represented by the symbol AÃ‡B.

For example, let A = {a, b, c, d} and let B = {c, d, e}. Then A Ã‡ B = {c, d}.

The intersection of two sets is defined in symbols as follows:

**A ****Ã‡**** B = {x: x is in A and x is in B}.**** **

### EXERCISE

Give the intersections of the following pairs of sets.

## A = {2, 4, 6, 8, 10, 12}

B = {3, 6, 9, 12}.

## A = the time from 6:00 a.m. to 6:00 p.m.

B = the time from 12:00 noon to 12:00 midnight

## __Commutative Law for ____Unions__In the union of two sets it does not matter which set is written first. For example,

{a, b, c} Ãˆ {c, d} = {a, b, c, d}

and

{c, d} Ãˆ {a, b, c} = {a, b, c, d}.

This may also be seen in the definitions

A Ãˆ B = {x: x is in A or x is in B}

and

B Ãˆ A = {x: x is in B or x is in A}

because, by the commutative law for the logical OR, *x is in A or x is in B* has the same meaning as *x is in B or x is in A*.

This fact is called the commutative law for the union of sets. It is summarized in symbols as follows:

**A ****Ãˆ**** B = B ****Ãˆ**** A.**** **

__Associative Law for Unions__

The union of A Ãˆ B with another set C is a composite union:

(A + B) + C = {x: (x is in A or x is in B) or x is in C}.

By the associative law for the logical OR, this is equal to:

{x: x is in A or (x is in B or x is in C)} = A + (B + C).

Therefore we have the associative law for unions, which is summarized as follows:

**(A ****Ãˆ**** B) ****Ãˆ**** C = A ****Ãˆ**** (B ****Ãˆ**** C).**** **

As a result of this law we may omit the brackets and write:

A + B + C.

For example, let A be the set of all African people, let B be the set of all Asian people, and let C be the set of all European people. Then the union A + B + C is the set of all the people in Africa, Asia, and

** **__Commutative Law for Intersections__** **** **

In the intersection of two sets it does not matter which set is written first. For example,

{a, b, c, d} Ã‡ {c, d, e} = {c, d}

and

{c, d, e} Ã‡ {a, b, c, d} = {c, d}.

This may also be seen in the definitions

A Ã‡ B = {x: x is in A and x is in B}

and

B Ã‡ A = {x: x is in B and x is in A}

because, by the commutative law for the logical AND, *x is in A and x is in B* has the same meaning as *x is in B and x is in A*.This fact is called the commutative law for the intersection of two sets. It is summarized in symbols as follows:

**A ****Ã‡**** B = B ****Ã‡**** A.**** **

__Associative Law for Intersections__

The intersection of A Ã‡ B with another set C is a composite intersection:

(A Ã‡ B) Ã‡ C = {x: (x is in A and x is in B) and x is in C}.

By the associative law for the logical AND, this is equal to

{x: x is in A and (x is in B and x is in C)} = A Ã‡ (B Ã‡ C).

Therefore we have the associative law for intersections, which is summarized in symbols as follows:

**(A ****Ã‡**** B) ****Ã‡**** C = A ****Ã‡**** (B ****Ã‡**** C)**** **

As a result of this law we may omit the brackets and write

A Ã‡ B Ã‡ C

For example, let A be the set of all African people, let B be the set of all female people, and let C be the set of all children under 10 years old. Then A . B . C is the set of all African girls under 10 years old.

__Distributive Law:__

Union over Intersection

The union of a set A with an intersection B . C is a composite expression:

A + (B . C) = {x: x is in A or (x is in B and x is in C)}.

By one of the distributive laws for composite statements, *x is in A or (x is in B and x is in C)* has the same meaning as *(x is in A or x is in B) and (x is in A or x is in C)*. Therefore

A + (B . C) = {x: (x is in A or x is in B) and (x is in A or x is in C)}.

But

{x: (x is in A or x is in B) and (x is in A or x is in C)} = (A + B) . (A + C).

Therefore

**A ****Ãˆ**** (B****Ã‡****C) = (A ****Ãˆ**** B) ****Ã‡**** (A ****Ãˆ**** C).**** **

This is one of the distributive laws for sets.

For example, suppose that applicants for a job must have either a university degree, or five years of work experience and a certificate of English language ability. Let A be the set of people with a degree, let B be the set of people with five years of experience, and let C be the set of people with a certificate of English. Then A + (B . C) is the set of people who may apply for the job. By the distributive law this is the same as (A + B) . (A + C). This shows that the applications may be checked in two separate ways before being accepted. One check makes sure that the applicant belongs to A + B (has a degree or five years of experience). The other check makes sure that the applicant belongs to A + C (has a degree or a certificate of English).

### EXERCISE

Suppose a person has enough money to buy a small new motor-car or a large second-hand motor-car. Let A be the set of small cars, let B be the set of large cars, and let C be the set of second-hand cars. Use the distributive law to find another way of saying that this person may buy a car which is small or large, and small or second-hand.

__Distributive Law: Intersection over__

Union

The intersection of a set A with a union B + C is a composite expression:

A . (B + C) = {x: x is in A and (x is in B or x is in C)}.

By one of the distributive laws for composite statements, *x is in A and (x is in B or x is in C)* has the same meaning as *(x is in A and x is in B) or (x is in A and x is in C)*. Therefore

A . (B + C) = {x: (x is in A and x is in B) or (x is in A and x is in C)}.

But {x: (x is in A and x is in B) or (x is in A and x is in C)} = (A . B) + (A . C).

Therefore

**A ****Ã‡**** (B ****Ãˆ**** C) = (A ****Ã‡**** B) ****Ãˆ**** (A ****Ã‡**** C).**** **

This is another distributive law for sets.

For example, let A be the set of children from 5 to 12 years old, let B be the set of boys, and let C be the set of girls. Then A . (B + C) means: the set of children who are 5 to 12 years old and either boys or girls. This is the same as the set (A . B) + (A . C), which means: children who are boys 5 to 12 years old or girls 5 to 12 years old.

### EXERCISE

Let A be the set of windy days in a particular year at a particular place, let B be the set of sunny days, and let C be the set of rainy days. Give the meaning of the two composite expressions:

## A . (B + C)

## (A . B) + (A . C).

__Complement__

The complement of a set A is the set of all elements in the universal set that do not belong to A. The complement of A is represented by the symbol A'. It is defined in symbols as follows:

A' = {x: x is not in A}.

For example, let the universal set be the set of whole numbers from 1 to 10, and let A be the set of all even numbers from 2 to 10. That is:

A = {2, 4, 6, 8, 10}.

Then A' is the set of all odd numbers from 1 to 9:

A' = {1, 3, 5, 7, 9}.