Section 1.5: Misc Exercises

Show that the function \(f : \mathbb{R} \rightarrow \{x \in \mathbb{R} : -1 < x < 1\}\) defined by
\[ f(x) = \frac{-x}{1 + |x|},\quad x \in \mathbb{R} \]
is one-one and onto function.

Show that the function \(f : \mathbb{R} \rightarrow \mathbb{R}\) given by \(f(x) = x^3\) is 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 \(P(X)\), \(A R B\) if and only if \(A \subseteq B\).
Is \(R\) an equivalence relation on \(P(X)\)? Justify your answer.

Find the number of all onto functions from the set \(\{1, 2, 3, \ldots, n\}\) to itself.

Let \(A = \{-1, 0, 1, 2\}\), \(B = \{-4, -2, 0, 2\}\) and \(f, g : A \rightarrow B\) be functions defined by
\[ f(x) = x^2 - x,\quad x \in A \]
and
\[ g(x) = 2\left\lfloor x - \frac{1}{2} \right\rfloor,\quad x \in A. \]
Are \(f\) and \(g\) equal?
Justify your answer. (Hint: One may note that two functions \(f : A \rightarrow B\) and \(g : A \rightarrow B\) such that \(f(a) = g(a)\ \forall\ a \in A\), are called equal functions).

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