On the set Z of all integer numbers- define the relation R by a R b iff a -2 b is divided by 3 - Then the relation R isA- only reflexiveB- reflexive and transitiveC- equivalence relationD- reflexive and symmetric

If a is an integer, then a+2a=3a; which is divisible by 3. ⇒ a R a ∀ a∈ℤ ∴ R is reflexive. Let a R b where a,b∈ℤ ⇒ a+2b=3k, nbsp;where k∈ℤ Now, b+2a=(3a+3b) ...

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Standard XI

Mathematics

Permutation: n Different Things Taken r at a Time

On the set Z ...

Question

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

only reflexive

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reflexive and transitive

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reflexive and symmetric

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equivalence relation

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Solution

The correct option is D equivalence relation
If $a$ is an integer, then $a + 2 a = 3 a;$ which is divisible by $3$ .
$\Rightarrow a R a \forall a \in Z$
$∴ R$ is reflexive.

Let $a R b$ where $a, b \in Z$
$\Rightarrow a + 2 b = 3 k,$ where $k \in Z$
Now,
$b + 2 a = (3 a + 3 b) - (a + 2 b)$
$= (3 a + 3 b) - 3 k$
$= 3 (a + b - k)$
$\Rightarrow b + 2 a = 3 m,$ where $m \in Z$
So, $a R b \Rightarrow b R a$
$∴ R$ is symmetric.

Again, let $a R b$ and $b R c$ where $a, b, c \in Z$
$\Rightarrow a + 2 b = 3 j$ and $b + 2 c = 3 k$
Now,
$a + 2 b = 3 j$
$a + 2 (3 k - 2 c) = 3 j$
$a + 2 c = 3 j - 6 k + 6 c$
$a + 2 c = 3 (j - 2 k + 2 c)$
$a + 2 c = 3 m,$ where $m \in Z$ as $j, k, c \in Z$
So, $a R b, b R c \Rightarrow a R c$
$∴ R$ is transitive.

Hence, $R$ is an equivalence relation.

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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

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