X and Y are two sets and f : X Y . If { f ( c ) = y ; c X , y Y } and { f - 1 ( d ) = x ; d Y , x X } , then the true statement is [2005]

Question

$X$ and $Y$ are two sets and $f : X \to Y$ . If ${f (c) = y; c \subset X, y \subset Y}$ and ${f^{- 1} (d) = x; d \subset Y, x \subset X},$ then the true statement is [2005]

Smart Achievers · Accepted Answer

(4)

Given that $X$ and $Y$ are two sets and $f : X \to Y$ .

${f (c) = y; c \subset X, y \subset Y}$ and ${f^{- 1} (d) = x; d \subset Y, x \subset X}$

The pictorial representation of the given information is as shown:

$Since f^{- 1} (d) = x \Rightarrow f (x) = d$

Now if $a \subset x \Rightarrow f (a) \subset f (x) = d \Rightarrow f^{- 1} (f (a)) = a$

Hence, $f^{- 1} (f (a)) = a,$ $a \subset x$ is the correct option.

Solution

Q.
$X$ and $Y$ are two sets and $f : X \to Y$ . If ${f (c) = y; c \subset X, y \subset Y}$ and ${f^{- 1} (d) = x; d \subset Y, x \subset X},$ then the true statement is [2005]

1 $f (f^{- 1} (b)) = b$

2 $f^{- 1} (f (a)) = a$

3 $f (f^{- 1} (b)) = b, b \subset Y$

4 $f^{- 1} (f (a)) = a, a \subset X$

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Solution

Q. X and Y are two sets and f:X→Y. If {f(c)=y; c⊂X, y⊂Y} and {f-1(d)=x; d⊂Y, x⊂X}, then the true statement is [2005]

1 f(f-1(b))=b

2 f-1(f(a))=a

3 f(f-1(b))=b, b⊂Y

4 f-1(f(a))=a, a⊂X