Q). Explain note
Note: The total number of relations that can be defined from a set A to a set B is the number of possible subsets of $A\times B$. If n(A) = p and n(B) = q, then $n\left(A\times B\right)=pq$ and the total number of relations is $2^{pq}$.

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$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{if} \mathrm{A} \mathrm{is} \mathrm{a} \mathrm{set} \mathrm{having} \text{'}\mathrm{n}\text{'} \mathrm{elements}, \mathrm{then} \\ \mathrm{Number} \mathrm{of} \mathrm{subsets} \mathrm{of} \mathrm{A}=2^{\mathrm{n}} \mathrm{i}.\mathrm{e}. 2^{\mathrm{Number} \mathrm{of} \mathrm{elements}} \\ \mathrm{Now} \mathrm{every} \mathrm{relation} \mathrm{from} \mathrm{set} \mathrm{A}\hspace{0.17em}\mathrm{to} \mathrm{set} \mathrm{B} \mathrm{is} \mathrm{subset} \mathrm{of} \mathrm{A}\times \mathrm{B}. \\ \mathrm{Also} \mathrm{if} \mathrm{number} \mathrm{of} \mathrm{elements} \mathrm{in} \mathrm{set} \mathrm{A} \mathrm{is} \mathrm{p} , \mathrm{number} \mathrm{of} \mathrm{elements} \\ \mathrm{in} \mathrm{set} \mathrm{B} \mathrm{is} \mathrm{q}, \mathrm{then} \mathrm{number} \mathrm{of} \mathrm{elements} \mathrm{in} \mathrm{A}\times \mathrm{B} \mathrm{is} \mathrm{pq} \mathrm{i}.\mathrm{e}. \\ \mathrm{n}\left(\mathrm{A}\right)=\mathrm{p} \\ \mathrm{n}\left(\mathrm{B}\right)=\mathrm{q} \\ \mathrm{n}\left(\mathrm{A}\times \mathrm{B}\right)=\mathrm{pq} \\ \mathrm{Hence} \\ \mathrm{Number} \mathrm{of} \mathrm{relations}=\mathrm{Number} \mathrm{of} \mathrm{subsets} \mathrm{of} \mathrm{A}\times \mathrm{B} \\ =2^{\mathrm{Number} \mathrm{of} \mathrm{elements} \mathrm{in} \mathrm{A}\times \mathrm{B}} \\ =2^{\mathrm{pq}} \end{gathered}$$

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