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

Period of $f(x) = nx + n – [nx + n]$, $n \in N$

where [ ] denotes the greatest integer function is :

normal

Let $\phi (x) = (x) + {2^{\log _x^3}} – {3^{\log _x^2}}$ then

normal

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements.

$I.$ $f$ is an odd function.

$II.$ $f$ is an even function.

$III$. $f$ is differentiable everywhere. Then,

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(KVPY-2019)

Which one of the following is not bounded on the intervals as indicated

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normal

(JEE MAIN-2022)

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

Solution

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Period of $f(x) = nx + n – [nx + n]$, $n \in N$

where [ ] denotes the greatest integer function is :

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements.

$I.$ $f$ is an odd function.

$II.$ $f$ is an even function.

$III$. $f$ is differentiable everywhere. Then,