Show that the function \( f : \mathbb{R} \to \{x \,\in\, \mathbb{R} : -1 < x < 1\} \) defined by \( f(x) = \dfrac{x}{1 + |x|} \), \( x \in \mathbb{R} \), is one-one and onto.
No.
Show that the function \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x) = x^3 \) is injective.
Injective.
Given a non-empty set \( X \), consider \( P(X) \) which is the set of all subsets of \( X \). Define the relation \( R \) in \( P(X) \) as follows:
For subsets \( A, B \) in \( P(X) \), \( ARB \) if and only if \( A \subseteq B \). Is \( R \) an equivalence relation on \( P(X) \)? Justify your answer.
No.
Find the number of all onto functions from the set \( \{1,2,3,\ldots,n\} \) to itself.
\( n! \)
Let \( A = \{-1, 0, 1, 2\} \), \( B = \{-4, -2, 0, 2\} \) and let \( f, g : A \to B \) be functions defined by
\( f(x) = x^2 - x, \; x \in A \)
and
\( g(x) = 2\left|x - \dfrac{1}{2}\right| - 1, \; x \in A \).
Are \( f \) and \( g \) equal? Justify your answer.
Yes.
Let \( A = \{1,2,3\} \). Then number of relations containing \((1,2)\) and \((1,3)\) which are reflexive and symmetric but not transitive is:
(A) 1 (B) 2 (C) 3 (D) 4
A
Let \( A = \{1,2,3\} \). Then number of equivalence relations containing \((1,2)\) is:
(A) 1 (B) 2 (C) 3 (D) 4
B