Operation of sets I

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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 È B.

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

A È 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 AÇB.

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

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

A Ç 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 UnionsIn the union of two sets it does not matter which set is written first. For example,

{a, b, c} È {c, d} = {a, b, c, d}

and

{c, d} È {a, b, c} = {a, b, c, d}.

This may also be seen in the definitions

A È B = {x: x is in A or x is in B}

and

B È 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 È B = B È A.

Associative Law for Unions

The union of A È 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 È B) È C = A È (B È 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
Europe.

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} Ç {c, d, e} = {c, d}

and

{c, d, e} Ç {a, b, c, d} = {c, d}.

This may also be seen in the definitions

A Ç B = {x: x is in A and x is in B}

and

B Ç 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 Ç B = B Ç A.

Associative Law for Intersections

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

(A Ç B) Ç 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 Ç (B Ç C).

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

(A Ç B) Ç C = A Ç (B Ç C)

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

A Ç B Ç 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 È (BÇC) = (A È B) Ç (A È 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 Ç (B È C) = (A Ç B) È (A Ç 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}.

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