Relation · a ≤ b²
Check whether the relation S in the set of all real numbers (R) defined by
S = {(a, b) : a ≤ b²} is reflexive, symmetric or transitive. Also, determine all x ∈ R such that (x, x) ∈ S.
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Reflexive :
(½, ½) ∉ S because ½ ≤ (½)² is false. ∴ not reflexive.
Symmetric :
(–1, 3) ∈ S but (3, –1) ∉ S. ∴ not symmetric.
Transitive :
(1, –2) ∈ S and (–2, 0) ∈ S but (1, 0) ∉ S. ∴ not transitive.
(x, x) ∈ S :
x ≤ x² ⇒ x² – x ≥ 0 ⇒ x(x–1) ≥ 0 ⇒ x ∈ (–∞, 0] ∪ [1, ∞).
i.e. x ∈ R – (0, 1).
Relation · a ≤ b³
Check whether the relation S in the set of all real numbers (R) defined by
S = {(a, b) : a ≤ b³} is reflexive, symmetric or transitive.
📖 Show Answer
Reflexive :
(½, ½) ∉ S because ½ ≤ (½)³ is false. ∴ not reflexive.
Symmetric :
(–1, 3) ∈ S but (3, –1) ∉ S. ∴ not symmetric.
Transitive :
(3, 3/2) ∈ S and (3/2, 4/3) ∈ S but (3, 4/3) ∉ S. ∴ not transitive.
Equivalence Relation · 2 Divides (a+b)
Prove that the relation R in the set of integers Z defined as
R = {(a, b): 2 divides (a+b)} is an equivalence relation. Also, determine [3].
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Reflexive :
a+a = 2a is divisible by 2 ⇒ (a,a) ∈ R. ∴ reflexive.
Symmetric :
If 2 divides a+b, then 2 divides b+a ⇒ (b,a) ∈ R. ∴ symmetric.
Transitive :
If a+b and b+c are even, then a and b have same parity, b and c have same parity ⇒ a and c have same parity ⇒ a+c is even. ∴ transitive.
[3] :
[3] = {…, –3, –1, 1, 3, 5, …} i.e. all odd integers.