Operation of sets II

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EXERCISE

Let the universal set be the set of all people. Give the complement of the set of all people who can swim.

Complement of a Complement

The complement of a complement is written in symbols as follows: (A')' = {x: not-(x is not in A)}.

Since not-(x is not in A) has the same meaning as x is in A, it follows that (A')' = {x: x is in A}. But {x: x is in A} = A. Therefore we have the complement law:

(A')' = A.

For example, let the universal set be the set of whole numbers from 1 to 10, and let A be the set of even numbers from 2 to 10. Then A' is the set of odd numbers from 1 to 9, and (A')' is the original set A of even numbers from 2 to 10.

EXERCISE

Let the universal set consist of all the books in a library. Let A be the set of books written in English. What are the sets A' and (A')'?

Union of a Set and its Complement

The union of a set and its complement is the universal set.

For example, let x stand for an animal, let A = {x: x is male}, and let A' = {x: x is female}. Then A + A' = {x: x is male or x is female} = the set of all animals.

We write this law briefly as follows:

A È A' = U.

EXERCISE

Give the universal sets for the following sets and their complements:

  1. A = all buildings with only one floor.
    A' = all buildings with two or more floors.
  2. A = all students who passed an examination
    A' = all students who failed the examination.

Intersection of a Set and its Complement

The intersection of a set and its complement is the empty set.

For example, let A consist of all the unbroken plates in a set of plates, and let A' consist of all the broken plates. Then A . A' is empty because there are no plates that are unbroken and broken at the same time.

We summarize this law in symbols as follows:

A Ç A' = O.

EXERCISE

Say whether or not the intersection of the following pairs of sets is the empty set:

  1. A = all sunny days in a year
    B = all windy days in the year.
  2. A = all wet days in a year
    B = all dry days in the year.

Complement of a Union

Since the union of two sets A and B is given by

A + B = {x: x is in A or x is in B},

the complement of the union is given by

(A + B)' = {x: not-(x is in A or x is in B)}

But not-(x is in A or x is in B) has the same meaning as x is not in A and x is not in B, and

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

Therefore, we have the following law for the complement of a union:

(A È B)' = A' Ç B'.

This is one of De Morgan's laws for sets.

For example, let the universal set be the set of all substances. Let A be the set of all solids, such as stone and iron. Let B be the set of all liquids, such as water and oil. Then A + B is the set all substances that are solid or liquid. The complement (A + B)' is the set of all substances that are not solid or liquid, in other words the set A' . B' of all substances that are not solid and not liquid, such as oxygen and nitrogen (which are gases).

EXERCISE

at A = all women and girls, let B = all children (girls and boys). Say what people are members of the complement of A + B.

Complement of an Intersection

Since the intersection of two sets A and B is given by

A . B = {x: x is in A and x is in B},

the complement of this intersection is given by

(A . B)' = {x: not-(x is in A and x is in B)}.

But not-(x is in A and x is in B) has the same meaning as x is not in A or x is not in B, and

{x: x is not in A or x is not in B} = A' + B'.

Therefore, we have the following law for the complement of an intersection:

(A Ç B)' = A' È B'.

This is another of De Morgan's laws for sets.

For example, let the universal set be the whole numbers from 1 to 10. Let A = {1, 2, 3, 4, 7, 8, 9, 10} and let B = {2, 4, 6, 8, 10}. Then A . B = {2, 4, 8, 10}, and so (A . B)' = {1, 3, 5, 6, 7, 9}. Also A' = {5, 6}, B' = {1, 3, 5, 7, 9} and A' + B' = {1, 3, 5, 6, 7, 9}. Therefore, in this example, (A . B)' = A' + B'.

EXERCISE

Let A = all women and girls, and let B = all children (girls and boys). Say what people are members of the complement of A . B when the universal set is all people.

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