If x and y are two sets and x' denotes the complement of x, then x intersection ( x union y) ' is equal to
Answers
Keep in mind that sets are mainly denoted by capital letters!
We're given two sets X and Y. X' represents the compliment of the set X, which includes the elements that are not in X but in universal set, U.
Likewise, Y' represents the compliment of Y having elements not in the set Y but in U.
It's just X' = U - X and Y' = U - Y.
By the way, what about (X ∪ Y)' ?!
Consider an element a ∈ (X ∪ Y)'.
We know the definition of the union of two sets. X ∪ Y contains the elements which are either in X or in Y.
But X ∪ Y does not contain elements which are not in both X and Y (not common elements are mentioned here). So,
a ∈ (X ∪ Y)' ⇒ a ∉ (X ∪ Y) ⇒ a ∉ X and a ∉ Y
If an element is not in a set, then it is present outside the set but in the universal set, i.e., that element is included in the compliment of that set.
So we can say that,
a ∉ X and a ∉ Y ⇒ a ∈ X' and a ∈ Y'
The elements seen common in two sets are also included in the intersection of that two sets. So,
a ∈ X' and a ∈ Y' ⇒ a ∈ (X' ∩ Y')
Thus we conclude with the relation:
(X ∪ Y)' = X' ∩ Y'
Likewise,
(X ∩ Y)' = X' ∪ Y'
These two laws are known as De Morgan's Laws. The laws state that the compliment of the union and intersection of two sets is respectively the intersection and the union of their compliments.
Okay, let's come to the question.
X ∩ (X ∪ Y)'
On applying De Morgan's Law, we get,
X ∩ (X' ∩ Y')
By the associative law for the intersection of sets, we have,
(X ∩ X') ∩ Y'
The intersection of a set and its compliment is always a null set, since both are disjoint. So X ∩ X' = ∅ :-
∅ ∩ Y'
Since a null set is a subset of every set, the intersection of a set with the null set is always a null set.
[Just remember A ∩ B = A ⇔ A ⊆ B]
So,
∅ ∩ Y' = ∅
Hence the answer is silly a null set!
Answer:
phi or null set
Explanation:
For explanation, please refer to the given attachment.
Hope it will help you.
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