(C) By definition, $x \in f^{-1}(A - B)$ if and only if $f(x) \in A - B$.This means $f(x) \in A$ and $f(x) \notin B$.This is equivalent to $x \in f^{-

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$f^{-1}(A - B) = f^{-1}(A) - f^{-1}(B)$

(C) By definition, $x \in f^{-1}(A - B)$ if and only if $f(x) \in A - B$.This means $f(x) \in A$ and $f(x) \notin B$.This is equivalent to $x \in f^{-1}(A)$ and $x \notin f^{-1}(B)$.Therefore, $x \in f^{-1}(A) - f^{-1}(B)$.Since this holds in both directions, we have $f^{-1}(A - B) = f^{-1}(A) - f^{-1}(B)$.

Class 12 Mathematics Relation and Function Mix Examples of Relation and Function

Easy

Let $f: X \rightarrow Y$ be a function and $A, B$ be non-void subsets of $Y$. Which of the following is true?

WBJEE 2019 All WBJEE

A
$f^{-1}(A) - f^{-1}(B) \supset f^{-1}(A - B)$ but the opposite does not hold.
B
$f^{-1}(A) - f^{-1}(B) \subset f^{-1}(A - B)$ but the opposite does not hold.
C
$f^{-1}(A - B) = f^{-1}(A) - f^{-1}(B)$
D
$f^{-1}(A - B) = f^{-1}(A) \cup f^{-1}(B)$

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Mix Examples of Relation and Function

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Let $f(x)$ and $g(x)$ be two continuous functions defined from $R \rightarrow R$,such that $f(x_1) > f(x_2)$ and $g(x_1) < g(x_2)$ for all $x_1 > x_2$. Then the solution set of $f(g(\alpha^2 - 2\alpha)) > f(g(3\alpha - 4))$ is

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Let $f(x) = x^2, x \in R$. For any $A \subseteq R$,define $g(A) = \{x \in R : f(x) \in A\}$. If $S = [0, 4]$,then which one of the following statements is not true?

DifficultJEE MAIN 2019

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Let $f(x) = x^{12} - x^9 + x^4 - x + 1$. Which of the following is true?

DifficultKVPY 2011

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Let $A = \{1, 3, 4, 6, 9\}$ and $B = \{2, 4, 5, 8, 10\}$. Let $R$ be a relation defined on $A \times B$ such that $R = \{((a_1, b_1), (a_2, b_2)) : a_1 \leq b_2 \text{ and } b_1 \leq a_2\}$. Then the number of elements in the set $R$ is

DifficultJEE MAIN 2023

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If $R \subset A \times B$ and $S \subset B \times C$,then the relation $(SoR)^{-1} = $

Medium

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Let $f: X \rightarrow Y$ be a function and $A, B$ be non-void subsets of $Y$. Which of the following is true?

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Let $f(x)$ and $g(x)$ be two continuous functions defined from $R \rightarrow R$,such that $f(x_1) > f(x_2)$ and $g(x_1) < g(x_2)$ for all $x_1 > x_2$. Then the solution set of $f(g(\alpha^2 - 2\alpha)) > f(g(3\alpha - 4))$ is

Let $f(x) = x^2, x \in R$. For any $A \subseteq R$,define $g(A) = \{x \in R : f(x) \in A\}$. If $S = [0, 4]$,then which one of the following statements is not true?

Let $f(x) = x^{12} - x^9 + x^4 - x + 1$. Which of the following is true?

Let $A = \{1, 3, 4, 6, 9\}$ and $B = \{2, 4, 5, 8, 10\}$. Let $R$ be a relation defined on $A \times B$ such that $R = \{((a_1, b_1), (a_2, b_2)) : a_1 \leq b_2 \text{ and } b_1 \leq a_2\}$. Then the number of elements in the set $R$ is

If $R \subset A \times B$ and $S \subset B \times C$,then the relation $(SoR)^{-1} = $

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