Show that the relation R defined by R = {(a, b): a – b is divisible by 3; a, b ∈ Z} is an equivalence relation. Watch Solution
Show that the relation R on the set Z of integers, given by R = {(a, b) : 2 divides a – b}, is an equivalence relation. Watch Solution
Prove that the relation R on Z defined by (a, b) ∈ R ⇔ a – b is divisible by 5 is an equivalence relation on Z. Watch Solution
Let n be a fixed positive integer. Define a relation R on Z as follows : (a, b) ∈ R ⇔ a – b is divisible by n. Show that R is an equivalence relation on Z. Watch Solution
Let Z be the set of integers. Show that the relation R = {(a, b) : a, b ∈ Z and a + b is even} is an equivalence relation on Z. Watch Solution
m is said to be related to n if m and n are integers and m – n is divisible by 13. Does this define an equivalence relation ? Watch Solution
Let R be a relation on the set A of ordered pairs of non-zero integers defined by (x, y) R (u, v) iff xv = yu. Show that R is an equivalence relation. Watch Solution
Show that the relation R on the set A = {x ∈ Z ; 0 ≤ x ≤ 12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1. Watch Solution
Let L be the set of all lines in XY-plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4. Watch Solution
Show that the relation R, defined on the set A of all polygons as R = {(P1, P2) : P1 and P2 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 ? Watch Solution
Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation. Watch Solution
Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other, and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}. Watch Solution
Let S be a relation on the set R of all real numbers defined by S = {(a, b) ∈ R × R : a^2 + b^2 = 1}. Prove that S is not an equivalence relation on R. Watch Solution
Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0 be defined as follows : (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0 Prove that R is an equivalence relation on Z × Z0 Watch Solution
If R and S are relations on a set A, then prove the following (i)R and S are symmetric ⇔ R ⋂ S, and R ⋃ S is symmetric (ii) R is reflexive, and S is any relation ⇔ R ⋃ S is reflexive. Watch Solution
If R and S are transitive relations on a set A, then prove that R ⋃ S may not be a transitive relation on A. Watch Solution
Let C be the set of all complex numbers and C0 be the set of all non-zero complex numbers. Let a relation R on C0 be defined as z1Rz2⇔z1−z2/ z1+z2 is real for all z1,z2∈C0 Show that R is an equivalence relation. Watch Solution
