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Question

On the set Z of all integer numbers, define the relation R by a R b iff a+2b is divided by 3. Then the relation R is

A
only reflexive
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B
reflexive and transitive
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C
reflexive and symmetric
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D
equivalence relation
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Solution

The correct option is D equivalence relation
If a is an integer, then a+2a=3a; which is divisible by 3.
a R a aZ
R is reflexive.

Let a R b where a,bZ
a+2b=3k, where kZ
Now,
b+2a=(3a+3b)(a+2b)
=(3a+3b)3k
=3(a+bk)
b+2a=3m, where mZ
So, a R bb R a
R is symmetric.

Again, let a R b and b R c where a,b,cZ
a+2b=3j and b+2c=3k
Now,
a+2b=3j
a+2(3k2c)=3j
a+2c=3j6k+6c
a+2c=3(j2k+2c)
a+2c=3m, where mZ as j,k,cZ
So, a R b, b R ca R c
R is transitive.

Hence, R is an equivalence relation.

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