Question

If A is the set of the divisors of the number 15, B is the set of prime numbers smaller than 10 and C is the set of even numbers smaller than 9, then \[\left( {A \cup C} \right) \cap B\] is the set
A. {1,3,5}
B. {1,2,3}
C. {2,3,5}
D. {2,5}

Answer 1

Hint: To solve this question, we need to know the basic theory related to the union, intersections and complementation of logical statements. And also, here we will use the basic terminology like divisors of the number, set of prime numbers and set of even numbers as discussed below.

Complete step-by-step answer:
As mentioned in the question, set A is the set of the divisors of the number 15.
And we know that A divisor, also called a factor, of a number is a number which divides. For integers, only positive divisors are usually considered, though obviously the negative of any positive divisor is itself a divisor.
A={1,3,5,15}
As mentioned in the question, B is the set of prime numbers smaller than 10.
B={2,3,5,7}
As mentioned in the question, C is the set of even numbers smaller than 9.
C={2,4,6,8}
Now, as we know, the union of two sets A and B is the set of elements which are in A, in B, or in both A and B. For example, if A = {a, b, c, d} and B = {a, p} then $A \cup B$ = {a,b.c,d,p}.
Similarly, here, ($A \cup C$)={1,2,3,4,5,6,8,15}
The intersection of two sets A and B, denoted by A ∩ B, is the set of all objects that are members of both the sets A and B. In symbols, that is, x is an element of the intersection A ∩ B, if and only if x is both an element of A and an element of B.
\[\left( {A \cup C} \right) \cap B\]={2,3,5}
Thus, \[\left( {A \cup C} \right) \cap B\] is equivalent to set {2,3,5}.

Therefore, option (C) is the correct answer.

Note: It’s necessary to keep in mind that the collection of well-defined distinct objects is known as a set. The word well-defined refers to a specific property which makes it easy to identify whether the given object belongs to the set or not. any one of the objects in a set is called a member or an element of the set. If a is an element of a set A we write a$ \in $A.