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....
If A = {1, 2, 3, 4} define relations on A which have properties of being
Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.
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),...
Check whether the relation R on R defined as R = {(a, b): a ≤ b3} is reflexive, symmetric or transitive.
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...
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.
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,...
The following relation is defined on the set of real numbers.
(i) aRb if a – b > 0
(ii) aRb iff 1 + a b > 0
(iii) aRb if |a| ≤ b. Find whether relation is reflexive, symmetric or transitive.
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...
Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R2 = {(2, 2), (3, 1), (1, 3)}, R3 = {(1, 3), (3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is
(i) reflexive
(ii) symmetric
(iii) transitive.
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...
Test whether the following relation R1, R2, and R3 are
(i) reflexive
(ii) symmetric and
(iii) transitive:
(i) R1 on Q0 defined by (a, b) ∈ R1 ⇔ a = 1/b.
(ii) R2 on Z defined by (a, b) ∈ R2 ⇔ |a – b| ≤ 5
(iii) R3 on R defined by (a, b) ∈ R3 ⇔ a2 – 4ab + 3b2 = 0.
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...
Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows: R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}. Find whether or not each of the relations R1, R2, R3, R4 on A is
(i) reflexive
(ii) symmetric and
(iii) transitive.
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...
Let A be the set of all human beings in a town at a particular time. Determine whether of the following relation is reflexive, symmetric and transitive:
(iii) R = {(x, y): x is wife of y}
(iv) R = {(x, y): x is father of y}
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:...
Let A be the set of all human beings in a town at a particular time. Determine whether of the following relation is reflexive, symmetric and transitive:
(i) R = {(x, y): x and y work at the same place}
(ii) R = {(x, y): x and y live in the same locality}
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 ⇒...