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CBSE 12 Math Relations and Functions Practice Paper 01
1. Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as R = {(x, y) : 3x – y = 0}
(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x}
(iv) Relation R in the set Z of all integers defined as R = {(x, y) : x – y is an integer}
(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}
(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:
(i) \(f : N \rightarrow N\) given by \(f(x) = x^2\)
(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 \(n\in N\).
10. Let \(f : \{1, 3, 4\} \rightarrow \{1, 2, 5\}\) and \(g : \{1, 2, 5\} \rightarrow \{1, 3\}\) be given by \(f = \{(1, 2), (3, 5), (4, 1)\}\) and \(g = \{(1, 3), (2, 3), (5, 1)\}\). Write down gof.
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