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Relations and Functions Miscellaneous Exercise on Chapter 1 Solutions
- Q1: Show that the function $f : R \to \{x \in R : – 1 < x < 1\}$ defined by $f(x) = \frac{x}{1 + |x|}$, $x \in R$ is one one and onto function.
- Q2: Show that the function $f : R \to R$ given by $f(x) = x^3$ is injective.
- Q3: 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 $A \subset B$. Is $R$ an equivalence relation on $P(X)$? Justify your answer.
- Q4: Find the number of all onto functions from the set $\{1, 2, 3,......, n\}$ to itself.
- Q5: Let $A = \{– 1, 0, 1, 2\}$, $B = \{– 4, – 2, 0, 2\}$ and $f, g : A \to B$ be functions defined by $f(x) = x^2 – x$, $x \in A$ and $g(x) = 2|\frac{x}{2} - 1| - 1$, $x \in A$. Are $f$ and $g$ equal? Justify your answer. (Hint: One may note that two functions $f : A \to B$ and $g : A \to B$ such that $f(a) = g(a) \forall a \in A$, are called equal functions).
- Q6: Let $A = \{1, 2, 3\}$. Then number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive is
- Q7: Let $A = \{1, 2, 3\}$. Then number of equivalence relations containing $(1, 2)$ is
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