Draw appropriate Venn diagram for each of the following:
(i) (A ∪ B)'
(ii) A' ∩ B'
(iii) (A ∩ B)'
(iv) A' ∪ B'
(i) (A ∪ B)'
(ii) A' ∩ B'
(iii) (A ∩ B)'
(iv) A' ∪ B'
Solution and Explanation
(i) (A∪B)'
(ii) A'∩B'
(iii) (A∩B)'
(iv) A' ∪ B'
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Concepts Used:
Complement of a Set
The complement of a set is described as A’ = {x: x ∈ U and x ∉ A}
where,
A’ stands for the complement.
Complement of Sets Properties:
1. Complement Laws: The union of a set A and its complement A’ allows 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’ cause the empty set ∅.
A ∩ A’ = ∅
For Example: If U = {11, 12 , 13 , 14 , 15 } and A = {11 , 12 , 13 } then A’ = {14 , 15}. From this it can be seen that
A ∪ A’ = U = { 11 , 12 , 13 , 14 , 15}
Also, A ∩ A’ = ∅
2. Law of Double Complementation: According to the 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 = {11 , 12 , 13 , 14 , 15} and A = {11 , 12, 13} then A’ ={14 , 15}. Now if we consider the complement of set ‘A’ we get,
(A’)’ = {11 , 12 , 13} = A
This gives back the set A itself.
3. 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’ = ∅
This law is accessible.