# Operation of sets II

### 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 ****Ãˆ ****A' = U. **

### EXERCISE

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

- A = all buildings with only one floor.

A' = all buildings with two or more floors. - 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 ****Ã‡ ****A' = O. **

### EXERCISE

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

- A = all sunny days in a year

B = all windy days in the year. - 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 ****Ãˆ ****B)' = A' ****Ã‡ ****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 ****Ã‡ ****B)' = A' ****Ãˆ ****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.