and 
is even 
. Show that 
is an equivalence relation on 
.Solution: $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathrm{Z}$ and $(\mathrm{a}+\mathrm{b})$ is even $\}$ (As given) If $R$ is Reflexive, Symmetric and Transitive, then $R$...
Solution: $a + 3b = 12$ (given) $a = 12 – 3b$ Putting $b = 1$ $a = 12 – 3(1) = 9$ Putting $b = 2$ $a = 12 – 3(2) = 6$ Putting $b = 3$ $a = 12 – 3(3) = 3$ Putting $b = 4$ $a = 12 – 3(4) = 0$; which...
Solution: Given set A = {x ∈ Z; 0 ≤ x ≤ 12} Also given that relation R = {(a, b): a = b} is defined on set A Now we have to check whether the given relation is equivalence or not. To prove...
Solution: First let R be a relation on A It is given that set A of ordered pair of integers defined by (x, y) R (u, v) if xv = y u Now we have to check whether the given relation is equivalence or...
Solution: Given that m is said to be related to n if m and n are integers and m − n is divisible by 13 Now we have to check whether the given relation is equivalence or not. To prove equivalence...
Solution: Given R = {(a, b): a, b ∈ Z and a + b is even} is a relation defined on R. Also given that Z be the set of integers To prove equivalence relation it is necessary that the given relation...
Solution: Given (a, b) ∈ R ⇔ a − b is divisible by n is a relation R defined on Z. To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and...
Solution: Given relation R on Z defined by (a, b) ∈ R ⇔ a − b is divisible by 5 To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive....
Solution: Given R = {(a, b): 2 divides a – b} is a relation defined on Z. To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive. Let us...
Solution: Given R = {(a, b): a – b is divisible by 3; a, b ∈ Z} is a relation To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive. Let...
Solution: (i) The relation on A having properties of being reflexive, transitive, but not symmetric is R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 1)} Relation R satisfies reflexivity and transitivity....
Solution: Let A be a set. Then, Identity relation IA=IA is reflexive, since (a, a) ∈ A ∀a The converse of it need not be necessarily true. Consider the set A = {1, 2, 3} Here, Relation R = {(1, 1),...
Solution: Given R = {(a, b): a ≤ b3} It is observed that (1/2, 1/2) in R as 1/2 > (1/2)3 = 1/8 ∴ R is not reflexive. Now, (1, 2) ∈ R (as 1 < 23 = 8) But, (2, 1) ∉ R (as 2 > 13 = 1) ∴ R is...
Solution: Given R = {(a, b): b = a + 1} Now for this relation we have to check whether it is reflexive, transitive and symmetric Reflexivity: Let a be an arbitrary element of R. Then,...
Solution: (i) Consider aRb if a – b > 0 Now for this relation we have to check whether it is reflexive, transitive and symmetric. Reflexivity: Let a be an arbitrary element of R. Then, a ∈ R...
Solution: Consider R1 Given R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)} Reflexivity: Here, (1, 1), (2, 2), (3, 3) ∈R So, R1 is reflexive. Symmetry: Here, (2, 1) ∈ R1, But (1, 2) ∉ R1...
Solution: (i) Given R1 on Q0 defined by (a, b) ∈ R1 ⇔ a = 1/b. Reflexivity: Let a be an arbitrary element of R1. Then, a ∈ R1 ⇒ a ≠1/a for all a ∈ Q0 So, R1 is not reflexive. Symmetry: Let...
Solution: (i) Consider R1 Given R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)} Now we have check R1 is reflexive, symmetric and transitive Reflexive: Given (a, a), (b, b) and...
Solution; (iii) Given R = {(x, y): x is wife of y} Now we have to check whether the relation R is reflexive, symmetric and transitive. First let us check whether the relation is reflexive:...
Solution: (i) Given R = {(x, y): x and y work at the same place} Now we have to check whether the relation is reflexive: Let x be an arbitrary element of R. Then, x ∈R ⇒...
Answers: (i) If we read the complete sentence it becomes x and the letter x comes before ‘e’ in the English alphabet. Thus, the letters before the letter ‘e’ are a,b,c,d. ∴ Roster form = {a,b,c,d}...