Question

Which of the following is correct?
A. \[A \cap \phi = A\]
B. \[A \cap \phi = \phi \]
C. \[A \cap \phi = U\]
D. \[A \cap \phi = A'\]

Answer 1

Hint: We know that the intersection of two sets is a set that contains the elements present in both sets. \[\phi \] denotes a set that contains no element. By using this concept we will solve the given question.
Complete step by step solution:
In the question, we have to find a set that is the intersection of set \[A\] and \[\phi \].
We know that there is no element in the set \[\phi \].
This implies there is no common element between set \[A\] and \[\phi \].
Thus we conclude that, the intersection set \[A\] and \[\phi \] is an empty set.
We know that the empty set is denoted by \[\phi \].
Therefore,
\[A \cap \phi = \phi \].
The correct option is option B.
Option A
We can get the intersection of set \[A\] and set \[B\] is equal to set \[A\], only either set \[A\] is a subset of set \[B\] or set\[A\] is equal to set \[B\].
So, option A. is an incorrect option.
Option C
\[U\] denotes the universal set. The intersection of two sets never is a universal set until both sets are a universal set.
Option D
\[A'\]denotes the set the compliment set of the set. Compliment of a set contains those elements which are not an element of the set.
It never be possible the intersection of set \[A\]with another set is a complement of set \[A\].

Hence option B is the correct option.

Note: Students are often confused with the intersection and union. They consider the intersection sign as a union. The symbol of union is \[ \cup \]. The union of set \[A\] and \[\phi \] is set \[A\] which is an incorrect solution.