If the value of n(Y)+n(Z) is k^2, then |k| is Let \(S=\ 1,2,3,4,5,6\ \) and \(X\) be the set of all relations \(R\) from \(S\) to \(S\) that satisfy both the following properties: i. \(R\) has exactly 6 elements. ii. For each \((a, b) R\), we have \(|a-b| 2\). Let \(Y=\ R X\) : The range of \(R\) has exactly one element \(\ \) and \(Z=\ R X: R\) is a function from \(S\) to \(S\ \). Let \(n(A)\) denote the number of elements in a set \(A\).
A. A
given |a-b| 2 so if i.e. Total elements in X is ^ 20 C _6 Now for n ( Y ), range of R has exactly one element i.e. second elements must be constant in R and since R must have 6 element so it is not possible to satisfy both condition so n ( Y )=0. aligned & for n(z) \\ & 1 3,4,5,6 \\ & 2 4,5,6 \\ & 3 1,5,6 \\ & 4 1,2,6 \\ & 5 1,2,3 \\ & 6 1,2,3,4 \\ & aligned no. of relation that are function will be = ^4 C _1 ^3 C _1 ^3 C _1 ^3 C _1 ^3 C _1 ^4 C _1 aligned & =(4 3 3)^2=k^2 \\ & i.e. k =36 aligned
Related: Mathematics — Sets and Relations · All PYQ Banks