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

Give an example of a relation. Which is (v) Symmetric and transitive but not reflexive.

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 $f(x) = 3 – 4x$

Show that the function $f : R \to R$ given by $f(x) = x^3$ is injective.

Check the injectivity and surjectivity of the following functions: (ii) $f : Z \to Z$ given by $f(x) = x^2$

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.

Show that the relation $R$ in the set $A$ of points in a plane given by $R = \{(P, Q) :$ distance of the point $P$ from the origin is same as the distance of the point $Q$ from the origin\}, is an equivalence relation. Further, show that the set of all points related to a point $P \neq (0, 0)$ is the circle passing through $P$ with origin as centre.

Show that the relation $R$ in $R$ defined as $R = \{(a, b) : a \leq b\}$, is reflexive and transitive but not symmetric.

Check the injectivity and surjectivity of the following functions: (i) $f : N \to N$ given by $f(x) = x^2$

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 $f(x) = 1 + x^2$

Let $f : R \to R$ be defined as $f(x) = 3x$. Choose the correct answer.

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 $R_{*}$?

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 (b) $R = \{(x, y) : x$ and $y$ live in the same locality\}

Let $A = \{1, 2, 3\}$. Then number of equivalence relations containing $(1, 2)$ is

Show that the relation $R$ defined in the set $A$ of all triangles as $R = \{(T_1, T_2) : T_1$ is similar to $T_2\}$, is equivalence relation. Consider three right angle triangles $T_1$ with sides 3, 4, 5, $T_2$ with sides 5, 12, 13 and $T_3$ with sides 6, 8, 10. Which triangles among $T_1$, $T_2$ and $T_3$ are related?

Check whether the relation $R$ in $R$ defined by $R = \{(a, b) : a \leq b^3\}$ is reflexive, symmetric or transitive.

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 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.

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.

Determine whether each of the following relations are reflexive, symmetric and transitive: (iii) Relation $R$ in the set $A = \{1, 2, 3, 4, 5, 6\}$ as $R = \{(x, y) : y$ is divisible by $x\}$