Question

The universal set for the set of rational numbers and the set of irrational numbers is denoted by which of the following?
(a) R
(b) W
(c) Z
(d) Q

Answer 1

Hint: First, start by defining a universal set. Next, use the very important fact that rational numbers and irrational numbers, both are a subset of real numbers. So, they can be represented by a single set instead of different sets.

Complete step-by-step answer:
In this question, we need to find the universal set for the set of rational numbers and the set of irrational numbers.
This basic set is called the Universal Set. The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc. For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set R of real numbers. If we expand the set of integers to include all decimal numbers, we form the set of real numbers. The set of reals is sometimes denoted by R.
The set of rational numbers or irrational numbers is a subset of the set of real numbers.
Then the rational numbers subsets of this set get in the universal subset of Real numbers as well as for irrational numbers as they are subsets of Real Numbers set.
Since, rational numbers and irrational numbers, both are a subset of real numbers. So, eventually, we need to find the universal set of real numbers which is denoted by R.
Hence, (a) is the correct answer.

Note: In this question, it is very important to know that rational numbers and irrational numbers, both are a subset of real numbers. So, eventually, we need to find the universal set of real numbers.The numbers in the real number system is divided into two groups.One group is called rational numbers and the other is called irrational numbers.The set of irrational numbers include those numbers that cannot be written as ratio of two integers,decimals number that are non-terminating and decimals that do not have repeating pattern of digits.For ex: $\pi$ , $\sqrt2$ are irrational numbers.The set of rational numbers includes natural numbers, whole numbers, integers, numbers that can be expressed as the ratio of two integers, decimal number that terminate and decimal numbers that have a repeating pattern of digits. For ex:$\dfrac{1}{2}$, $\sqrt{81}$ are rational numbers.