Let X be a non-empty set and let P(X) denote collection of all subsets of X . Define f: X × P(X) ?

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1.Relation and Function

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Let $X$ be a non-empty set and let $P(X)$ denote the collection of all subsets of $X$. Define $f: X \times P(X) \rightarrow R$ by $f(x, A)=\left\{\begin{array}{ll}1, & \text { if } x \in A \\ 0, & \text { if } x \notin A^*\end{array}\right.$ Then, $f(x, A \cup B)$ equals

$f(x, A)+f(x, B)$

$f(x, A)+f(x, B)-1$

$f(x, A)+|f(x, A)-f(x, B)|$

(KVPY-2011)

Solution

(c)

We have,

$\begin{array}{c}f: X \times P(X) \rightarrow R \\f(x, A)=\left\{\begin{array}{ll} 1, & \text { if } x \in A \\0, & \text { if } x \notin A\end{array}\right. \\f(x, A \cup B)=\left\{\begin{array}{ll}1, & \text { if } x \in A \cup B \\0, & \text { if } x \notin A \cup B\end{array}\right. \\\text { If } x \in A, x \in B \Rightarrow f(x, A \cup B)=1 \\\text { If } x \in A, x \notin B \Rightarrow f(x, A \cup B)=1 \\\text { If } x \notin A, x \in B \Rightarrow f(x, A \cup B)=1 \\ \text { If } x \notin A, x \notin B \Rightarrow f(x, A \cup B)=0\end{array}$

Standard 12

Mathematics