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Let R be a relation from set Q to Q def...

Let R be a relation from set Q to Q defined as:
`R={(a,b):a,b in Q and a -b in Z}`
Prove that
(i) For each `a in Q, (a,a) in R`
`(ii) (a,b) in R implies (b,c) in R` where `a, b in Q`
(iii) `(a,b) in R , (b,c) in R implies (a,c) in R ` , where `a, b ,c in Q`

Text Solution

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The correct Answer is:
N/a
(i) For each `a in Q`
`(a-a)=0 in Z`
`implies (a,a) in R`
`therefore ` For each `a in Q, (a,b) in R`
(ii) Let ` a, b in Q and (a,b) in R`
`implies a-b in Z`
`implies -(b-a)in Z`
`implies b-a in Z`
`implies (b,a) in R.` Hence Proved.
(iii) Let `a,b,c in Q and (a,b) in R and (b,c) in R`
`implies a-b in Z and b-c in Z`
`implies a-b+b-c in Z`
`implies a-c in Z `
`implies (a-c) in R`. Hence Proved.
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