Question

Let A be the set of all students of class XII and B be the set of all students of class XI of same school. Consider the following statements
I. $\\{(a,b)\in A\times B:a\text{ is the brother of }b\text{. }\\!\\!\\}\\!\\!\text{ }$
II. $\\{(a,b)\in B\times A:a\text{ is the sister of }b\text{. }\\!\\!\\}\\!\\!\text{ }$
III. $\\{(a,b)\in B\times A:\text{age of }a\text{ is greater than age of }b\text{. }\\!\\!\\}\\!\\!\text{ }$
IV. \\{(a,b)\in A\times B:Total\text{ marks obtained by }a\text{ in final examination is less than }
\text{total marks obtained by }b\text{ in final examination}\text{. }\\!\\!\\}\\!\\!\text{ }
V. $\\{(a,b)\in A\times B:a\text{ lives in the same locality as }b\text{. }\\!\\!\\}\\!\\!\text{ }$
Which of the following are relations from A to B.
(a) I, III, IV
(b) II, IV
(c) I, IV, V
(d) All of these

Answer 1

Hint : Think of the representations and definitions of the Cartesian product and the relation of two sets. Also, remember that relation and Cartesian product are directional quantities, i.e., the Cartesian product $A\times B\text{ and }B\times A$ are different. So, to find the statements which are relations from A to B, just check that (a,b) must belong to $A\times B$ and report the answer.

Complete step by step solution :
Before starting with the solution, we will discuss the relation and the Cartesian product of two sets. Let us start with the Cartesian product. The Cartesian product of two sets is defined as the set of all possible ordered pairs (a,b) such that a belongs to the first set and b belongs to the other one. This can be represented as $A\times B=\\{(a,b):a\in A\text{ and }b\in B\\}$ . Now if we talk about relation, a relation between two sets is a collection of ordered pairs such that the first term is from the first set and the second term is from the other set. A relation R from set A to B can be represented as: $R=\\{(a,b):a\in A\text{ and }b\in B,\text{ a and b are related }\\!\\!\\}\\!\\!\text{ }$ .
So, if we analyse the two definitions we will find that $A\times B$ and relation from A to B are related as: $R\subseteq A\times B$ .
So, using the above definitions, we can say that $\\{(a,b)\in A\times B:a\text{ is the brother of }b\text{. }\\!\\!\\}\\!\\!\text{ }$ , \\{(a,b)\in A\times B:Total\text{ marks obtained by }a\text{ in final examination is less than total marks } \text{obtained by }b\text{ in final examination}\text{. }\\!\\!\\}\\!\\!\text{ } and $\\{(a,b)\in A\times B:a\text{ lives in the same locality as }b\text{. }\\!\\!\\}\\!\\!\text{ }$ are relations from A to B. Hence, the correct answer is option (c).
Let us explore the other options as well. If we see option (a), It says that I, III and IV are relations from A to B, but III statement is not a relation from A to B, so option (a) is not the answer. Similarly, for option (b), statement II is not the relation from A to B.

Note : Remember that Cartesian product and relation are directional, i.e., $A\times B$ and $B\times A$ or relation from A to B and relation from B to A are not always equivalent, but are related as $R\left( A\text{ to }B \right)\subseteq A\times B$ . Also, remember that Cartesian product and relation are two very different quantities and signify different things, just the thing is they visually appear alike when written in the form of a set.