Question

If A is any set, then
A. $ A \cup A' = \phi $
B. $ A \cup A' = U $
C. $ A \cap A' = U $
D.None of these

Answer 1

Hint: In mathematics, the sets are one of the most fundamental concepts. Here we will use the concept of the set and its complement, union of two sets and the intersection of two sets and then will find the solution accordingly.

Complete step-by-step answer:
Set is defined as the collection of well-defined and distinct objects and considered as an object in its own right. It is the group of the things that belong together. Sets are denoted by capital alphabets such as A, B ,... and so on.
The complement of set A is denoted by A’ and it is the set of elements which belong to the Universal set but which do not belong to A.
Union of two sets – Let A and B be two different sets. Then the union of A and B is denoted by $ A \cup B $ . The union is the set that contains all the elements in A or in B or in both.
Here, we are asked to find the union of A and its complement –
I.e. $ A \cup A' $
Since by definition $ A' $ contains all the elements in the universal set except the elements of A.
Union of A and A’ is equal to all the elements in the Universal set.
 $ \therefore A \cup A' = U $
So, the correct answer is “Option B”.

Note: Know the difference between the union and intersection of two sets and apply accordingly. The intersection of A and B is a set where it contains elements common in A and B. Also, refer to the laws of the complement of the sets for better understanding.