Question

R={(a,b) : ab is a factor of 6} Determine whether the relation is reflexive, .symmetric or transitive.
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Answer 1

Given:

R={(a,b) : ab is a factor of 6}

To find:

Determine whether the relation is reflexive, .symmetric or transitive.

Solution:

First of all, let us write a few elements of Relation R.

R = {(1,6), (2,6), ....., (6,6) ...... , (2,3), (3,2), .....}

So, R is defined on a set A with values A = {1, 2, 3, 4, 5, 6,.......}

1. Checking whether it is symmetric:

A relation R is symmetric, if (a,b) is an element of R then (b,a) is also an element of R.

OR

$\forall (a,b) \in R \Rightarrow (b,a) \in R$

Let us consider element (2,3) product is 2 $\times$ 3 = 6

Now, Let us consider element (3,2) product is 3 $\times$ 2 = 6

It will be true for all the elements in relation R.

So, The relation R is symmetric.

2. Checking reflexive property:

A relation R is reflexive, if (a,a) is an element of R for all the elements of the domain set A.

OR

$(a,a) \in R\ \forall a \in A$

Let us consider element (1,1) product is 1 $\times$ 1 = 1

It does not satisfy the condition, so (1,1) can not be in R

So, The relation R is not reflexive.

3. Checking transitive property:

A relation R is transitive, if (a,b) and (b,c) are elements of R then (a,c) also an element of R.

OR

$\{(a,b), (b,c)\} \in R \Rightarrow (a,c) \in R$

Let us consider elements from the relation R : {(1,6), (6,2)}

Their product is 6 and 12 so both are in Relation R.

For this relation to be transitive, (1,2) should also be in R.

But product of 1 and 2 is 2 which does not satisfy the condition.

Hence, the condition $\{(a,b), (b,c)\} \in R \Rightarrow (a,c) \in R$ is not always true.

So, The relation R is not transitive.

The given relation is only Symmetric.

Answer 2

Answer:

Step-by-step explanation: