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Home » Let A = {a, b}. List all relations on A and find their number.
Let A = {a, b}. List all relations on A and find their number.
CBSE | Class 11 | Exercise 2.3 | Maths | RD Sharma | Relations
Let A = {a, b}. List all relations on A and find their number.

Solution:

The number of possible subsets of A × B is the total number of relations that can be defined from a set A to a set B.

If n (A) = p and n (B) = q, then we can write that:

n (A × B) = pq

Thus, the total number of relations is equal to 2pq.

Now we have,

    \[A\text{ }\times \text{ }A\text{ }=\text{ }\left\{ \left( a,\text{ }a \right),\text{ }\left( a,\text{ }b \right),\text{ }\left( b,\text{ }a \right),\text{ }\left( b,\text{ }b \right) \right\}\]

All possible subsets of A × A are given by the total number of relations. Therefore, we have:

[\left\{ \left( a,\text{ }a \right),\text{ }\left( a,\text{ }b \right),\text{ }\left( b,\text{ }a \right),\text{ }\left( b,\text{ }b \right) \right\},\text{ }

\left\{ \left( a,\text{ }a \right),\text{ }\left( a,\text{ }b \right) \right\},\text{ }\left\{ \left( a,\text{ }a \right),\text{ }\left( b,\text{ }a \right) \right\},

\left\{ \left( a,\text{ }a \right),\text{ }\left( b,\text{ }b \right) \right\},\text{ }\left\{ \left( a,\text{ }b \right),\text{ }\left( b,\text{ }a \right) \right\},\text{ }

\left\{ \left( a,\text{ }b \right),\text{ }\left( b,\text{ }b \right) \right\},\text{ }\left\{ \left( b,\text{ }a \right),\text{ }\left( b,\text{ }b \right) \right\},\text{ }

\left\{ \left( a,\text{ }a \right),\text{ }\left( a,\text{ }b \right),\text{ }\left( b,\text{ }a \right) \right\},\text{ }\left\{ \left( a,\text{ }b \right),\text{ }\left( b,\text{ }a \right),\text{ }\left( b,\text{ }b \right) \right\},\text{ }

\left\{ \left( a,\text{ }a \right),\text{ }\left( b,\text{ }a \right),\text{ }\left( b,\text{ }b \right) \right\},\text{ }\left\{ \left( a,\text{ }a \right),\text{ }\left( a,\text{ }b \right),\text{ }\left( b,\text{ }b \right) \right\},\text{ }

\left\{ \left( a,\text{ }a \right),\text{ }\left( a,\text{ }b \right),\text{ }\left( b,\text{ }a \right),\text{ }\left( b,\text{ }b \right) \right\}]

n (A) = 2

⇒ n (A × A) = 2 × 2

n (A × A)= 4

Therefore, the total number of relations = 24 = 16.

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