Let R be relation from Q to Q defined by R-a-b-a-b - Q and a-b - Z- Show that - a-b - R and b-c - R implies that a-c - R-

(a, b)∈ R⇒ a b∈ Z (b, c)∈ R⇒ b c∈ Z ⇒ (a c)=(a b)+(b c) being sum of two integers is also a integer ⇒ (a c)∈ z⇒ (a, c)∈ R . ...

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Definition of Relations

Let R be re...

Question

Let $R$ be relation from $Q$ to $Q$ defined by $R = {(a, b) : a, b \in Q a n d a - b \in Z}$ .
Show that :
$(a, b) \in R$ and $(b, c) \in R$ implies that $(a, c) \in R$ .

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Solution

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R = {(a, b) : a, b \in Q a n d a - b \in Z}

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Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b²}. Are the following true?

(i) (a, a) ∈ R, for all a ∈ N

(ii) (a, b) ∈ R, implies (b, a) ∈ R

(iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R.

Justify your answer in each case.

Let R be a relation from N to N defined by $R = {(a, b) : a, b ϵ N$ and $a = b^{2}$ }. Are the following true?

(i) $(a, a) ϵ R$ for all $a ϵ N$

(ii) $(a, a) ϵ R$ implies $(b, a) ϵ R$

(iii) $(a, b) ϵ R, (b, c) ϵ R$ implies $(a, c) ϵ R$

Justify your answer in each case.

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