(a)  R = {(x, y) : x and y work at the same place} 

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

(c)  R = {(x, y) : x is exactly 7 cm taller than y} (d)  R = {(x, y) : x is wife of y} 

(e)  R = {(x, y) : x is father of y} 

2. Show that the relation R in the set R of real numbers, defined as \(R = {(a, b) : a ≤ b2}\) is neither reflexive nor symmetric nor transitive. 

3. 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.

4. 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?

5. 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 \(y = 2x + 4\).

6. 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? 

7. Check the injectivity and surjectivity of the following functions: 

(ii) \(f : Z \rightarrow Z\) given by \(f(x) = x^2\) 

(iii) \(f : R \rightarrow R\) given by \(f(x) = x^2\) 

(iv) \(f : N \rightarrow N\) given by \(f(x) = x^3\) 

(v) \(f : Z \rightarrow Z\) given by \(f(x) = x^3\) 

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

9.  Let \(f : N \rightarrow N\) be defined by \(f(n) =\biggl\{\begin{matrix}\frac{n+1}{2}\text{  ,if n is odd}\\\frac{n}{2}\text{  ,if n is even}\end{matrix}\) for all \(n\in N\).