Complement of sets

Let’s say that we have a set A that is a subset of some universal set U. The complement of A is the set of elements of the universal set that are not elements of A. In our example above, the complement of {-2, -1, 0, 1} is the set containing all the integers that do not satisfy the inequality.
We can write A^c

You can also say complement of A in U

Example #1.

Let B = {1 orange, 1 pinapple, 1 banana, 1 apple}

Let U = {1 orange, 1 apricot, 1 pinapple, 1 banana, 1 mango, 1 apple, 1 kiwifruit }

Again, we show in bold all elements in U, but not in B

B^c = {1 apricot, 1 mango, 1 kiwifruit}

Example #2.

Find the complement of B in U

B = { 1, 2, 4, 6}

U = {1, 2, 4, 6, 7, 8, 9 }

Complement of B in U = { 7, 8, 9}

Some Properties of Complement Sets source

1) A ∪ A′ = U
2) A ∩ A′ = Φ
3) Law of double complement : (A′ )′ = A
4) Laws of empty set and universal set Φ′ = U and U′ = Φ.

Examples :

1) If A = { 1, 2, 3, 4} and U = { 1, 2, 3, 4, 5, 6, 7, 8} then find A complement ( A’).

Solution :
A = { 1, 2, 3, 4} and Universal set = U = { 1, 2, 3, 4, 5, 6, 7, 8}

Complement of set A contains the elements present in universal set but not in set A.

Elements are 5, 6, 7, 8.

∴ A complement = A’ = { 5, 6, 7, 8}.

2) If B = { x | x is a book on Algebra in your library} . Find B’.

Solution : B’ = { x | x is a book in your library and x ∉ B }

3) If A = { 1, 2, 3, 4, 5 } and U = N , then find A’.

Solution :
A = { 1, 2, 3, 4, 5 }

U = N

⇒ U = { 1, 2, 3, 4, 5, 6, 7, 8, 9,10,… }

A’ = { 6, 7, 8, 9, 10, … }

4) If A = { x | x is a multiple of 3, x ∉ N }. Find A’.

Solution :
As a convention, x ∉ N in the bracket indicates N is the universal set.

N = U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11, … }

A = { x | x is a multiple of 3, x ∉ N }

A = { 3, 6, 9, 12, 15, … }

So, A’ = { 1, 2, 4, 5, 7, 8, 10,11, … }