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?

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).

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\}$

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

Let $A = R – \{3\}$ and $B = R – \{1\}$. Consider the function $f : A \to B$ defined by $f(x) = \frac{x-2}{x-3}$. Is $f$ one-one and onto? Justify your answer.

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

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 $f(x) = 3 – 4x$

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

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.

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

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.

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 whether the relation $R$ defined in the set $\{1, 2, 3, 4, 5, 6\}$ as $R = \{(a, b) : b = a + 1\}$ is reflexive, symmetric or transitive.

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

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 $x$.

Show that the relation $R$ in the set $R$ of real numbers, defined as $R = \{(a, b) : a \leq b^2\}$ is neither reflexive nor symmetric nor transitive.

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

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

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