Give an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive.

Let $f : R \to R$ be defined as $f(x) = x^4$. Choose the correct answer.

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 $x < 4\}$

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

Give an example of a relation. Which is (iv) Reflexive and transitive but not symmetric.

Let $A = \{1, 2, 3\}$, $B = \{4, 5, 6, 7\}$ and let $f = \{(1, 4), (2, 5), (3, 6)\}$ be a function from $A$ to $B$. Show that $f$ is one-one.

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 (d) $R = \{(x, y) : x$ is wife of $y\}$

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

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 (e) $R = \{(x, y) : x$ is father of $y\}$

Give an example of a relation. Which is (iii) Reflexive and symmetric but not transitive.

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

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

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).

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

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 $f(x) = 3 – 4x$

Let $A = \{1, 2, 3\}$. Then number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive is

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 (c) $R = \{(x, y) : x$ is exactly 7 cm taller than $y\}$

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.

Prove that the Greatest Integer Function $f : R \to R$, given by $f(x) = [x]$, is neither one-one nor onto, where $[x]$ denotes the greatest integer less than or equal to $x$.