Show that the Modulus Function $f : R \to R$, given by $f(x) = |x|$, is neither one-one nor onto, where $|x|$ is $x$, if $x$ is positive or 0 and $|x|$ is $– x$, if $x$ is negative.

Show that the relation $R$ in the set $\{1, 2, 3\}$ given by $R = \{(1, 2), (2, 1)\}$ is symmetric but neither reflexive nor transitive.

Show that the function $f : R_{*} \to R_{*}$ defined by $f(x) = \frac{1}{x}$ is one-one and onto, where $R_{*}$ is the set of all non-zero real numbers. Is the result true, if the domain $R_{*}$ is replaced by $N$ with co-domain being same as $R_{*}$?

Let $A$ and $B$ be sets. Show that $f : A \times B \to B \times A$ such that $f(a, b) = (b, a)$ is bijective function.

Determine whether each of the following relations are reflexive, symmetric and transitive: (ii) Relation $R$ in the set $N$ of natural numbers defined as $R = \{(x, y) : y = x + 5$ and $x < 4\}$

Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (a) $R = \{(x, y) : x$ and $y$ work at the same place\}

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).

Show that the relation $R$ defined in the set $A$ of all polygons as $R = \{(P_1, P_2) : P_1$ and $P_2$ have same number of sides\}, is an equivalence relation. What is the set of all elements in $A$ related to the right angle triangle $T$ with sides 3, 4 and 5?

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) $f : R \to R$ defined by $f(x) = 3 – 4x$

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (ii) $f : R \to R$ defined by $f(x) = 1 + x^2$

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

Let $L$ be the set of all lines in XY plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1$ is parallel to $L_2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.

Show that the relation $R$ in the set $A$ of all the books in a library of a college, given by $R = \{(x, y) : x$ and $y$ have same number of pages\}$ is an equivalence relation.

Show that each of the relation $R$ in the set $A = \{x \in Z : 0 \leq x \leq 12\}$, given by (i) $R = \{(a, b) : |a – b|$ is a multiple of 4\} is an equivalence relation. Find the set of all elements related to 1 in each case.

Determine whether each of the following relations are reflexive, symmetric and transitive: (iv) Relation $R$ in the set $Z$ of all integers defined as $R = \{(x, y) : x – y$ is an integer\}$

Let $R$ be the relation in the set $\{1, 2, 3, 4\}$ given by $R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}$. Choose the correct answer.

Check the injectivity and surjectivity of the following functions: (iv) $f : N \to N$ given by $f(x) = x^3$

Show that the relation $R$ in the set $A = \{1, 2, 3, 4, 5\}$ given by $R = \{(a, b) : |a – b|$ is even\}, is an equivalence relation. Show that all the elements of $\{1, 3, 5\}$ are related to each other and all the elements of $\{2, 4\}$ are related to each other. But no element of $\{1, 3, 5\}$ is related to any element of $\{2, 4\}$.

Check the injectivity and surjectivity of the following functions: (v) $f : Z \to Z$ given by $f(x) = x^3$

Let $R$ be the relation in the set $N$ given by $R = \{(a, b) : a = b – 2, b > 6\}$. Choose the correct answer.