Question

Which of the following sets is a universal set for the other four sets?
A.The set of even natural numbers
B.The set of odd natural numbers
C.The set of natural numbers
D.The set of negative integers
E.The set of integers

Answer 1

Hint: A universal set should include all the elements of all other sets in it. Therefore, in this question, we should find out the elements in all the given sets and use their definition and then find out which set contains the elements of all other sets which will be our answer to the given question.

Complete step-by-step answer:
We know that the set of natural numbers is N={1,2,3…}, i.e. it can be defined as the set containing 1 and all the elements that can be obtained by adding a multiple of 1 to 1. For example, we can write the natural numbers 3 and 8 as $3=1+2\times 1,8=1+8\times 1$ and so on…………………. (1.1)
The set of integers (Z) contains 0, all the natural numbers and their negatives. Therefore, all the elements of natural numbers are also elements of the set of integers……………….(1.2)
The set of odd and even natural numbers contains the elements which are natural numbers, therefore, all the elements of this set are also elements of the set of natural numbers. However, from (1.2), we find that they should also be elements of the set of integers…………………………(1.3)
The set of negative integers contains integers which are negative, thus all the elements of the set of negative integers should also be the elements of the set of integers ……………………..(1.4)
From (1.1), (1.2), (1.3) and (1.4), we find that every element of any of the given sets should be an element of the set of integers and thus the set of integers is the universal set. Thus, (e) should be the correct answer.

Note: We should be careful to see that the universal set should contain all the elements of the given sets. However, it does not matter if the sets have any overlap among themselves, for example as the odd natural numbers are also natural numbers, if we take an odd natural number, it would belong to both the set of odd natural numbers and the set of natural numbers. However, Z can still be called the universal set as overlap among its constituent subsets does not matter in defining a universal set.