Question

properties of complement of a set

Answer 1

Properties of Complement of Sets:

i) Complement Laws: The union of a set A and its complement A’ gives the universal set U of which A and A’ are a subset.

A ∪ A’ = U

Also the intersection of a set A and its complement A’ gives the empty set ∅.

A ∩ A’ = ∅

For Example: If  U = {1 , 2 , 3 , 4 , 5 }  and  A = {1 , 2 , 3 } then  A’ = {4 , 5}. From this it can be seen that

A ∪ A’ = U = { 1 , 2 , 3 , 4 , 5}

Also

A ∩ A’ = ∅

ii) Law of Double Complementation:

According to this law if we take the complement of the complemented set A’ then, we get the set A itself.

(A’ )’ = A

In the previous example we can see that, if U = {1 , 2 , 3 , 4 , 5} and A = {1 , 2 ,3}  then A’ ={4 , 5} . Now if take the complement of set A’ we get,

(A’ )’ = {1 , 2 , 3} = A

This gives us the set A itself.

iii) Law of empty set and universal set:

According to this law the complement of the universal set gives us the empty set and vice-versa i.e.,

∅’ = U And U = ∅’