Discuss the following relations for reflexivity, symmetricity and transitivity:
(i) The relation R defined on the set of all positive integers by “mRn if m divides n”. Solution
(ii) Let P denote the set of all straight lines in a plane. The relation R defined by “lRm if l is perpendicular to m”.
(iii) Let A be the set consisting of all the members of a family. The relation R defined by “aRb if a is not a sister of b”. Solution
(iv) Let A be the set consisting of all the female members of a family. The relation R defined by “aRb if a is not a sister of b”. Solution
(v) On the set of natural numbers the relation R defined by “xRy if x + 2y = 1”. Solution
(2) Let X = {a, b, c, d} and R = {(a, a), (b, b), (a, c)}. Write down the minimum number of ordered pairs to be included to R to make it
(i) reflexive (ii) symmetric (iii) transitive (iv) equivalence
(3) Let A = {a, b, c} and R = {(a, a), (b, b), (a, c)}. Write down the minimum number of ordered pairs to be included to R to make it
(i) reflexive (ii) symmetric (iii) transitive (iv) equivalence
(4) Let P be the set of all triangles in a plane and R be the relation defined on P as aRb if a is similar to b. Prove that R is an equivalence relation Solution
(5) On the set of natural numbers let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is
(i) reflexive (ii) symmetric (iii) transitive (iv) equivalence
(6) Prove that the relation “friendship” is not an equivalence relation on the set of all people in Chennai.
(7) On the set of natural numbers let R be the relation defined by aRb if a + b ≤ 6. Write down the relation by listing all the pairs. Check whether it is
(i) reflexive (ii) symmetric (iii) transitive (iv) equivalence
(8) Let A = {a, b, c}. What is the equivalence relation of smallest cardinality on A? What is the equivalence relation of largest cardinality on A? Solution
(9) In the set Z of integers, define mRn if m − n is divisible by 7. Prove that R is an equivalence relation.
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