How many properties can be held by a group?

a)
2

b)
3

c)
4

d)
5

Correct answer is option 'C'. Can you explain this answer?

Question

How many properties can be held by a group?a)2b)3c)4d)5Correct answer is option 'C'. Can you explain this answer?

Accepted Answer

≠ R as every element is not related to itself.
≠ S as (1, 2) ∈ R but (2, 1) ∉ R
T (1,2), ∈ R, (2, 2) ∈ R
⇒ (1, 2)o (2, 2) = (1, 2) ∈ R

Answer

As (1,1) does not belongs to R , it is not reflexive.
As ( 1,2) belongs to R but ( 2,1) does not belongs to R
so it is not symmetric
but for all (a,b) and (b,c) there , (a,c) exists in R
so it is transitive

Answer

Explanation:

To determine whether the relation R is reflexive, symmetric, or transitive, we will define each of these properties and check if they hold for the given relation R.

Reflexive Property:
A relation R on a set A is reflexive if every element in A is related to itself.

In this case, the relation R={(2,2),(3,3),(4,4),(1,2)} contains the elements (2,2), (3,3), and (4,4), which indicate that each element is related to itself. However, the element (1,2) does not indicate that 1 is related to itself. Therefore, the relation R is not reflexive.

Symmetric Property:
A relation R on a set A is symmetric if for every pair (a,b) in R, the pair (b,a) is also in R.

In this case, the relation R={(2,2),(3,3),(4,4),(1,2)} contains the pairs (2,2), (3,3), and (4,4), which are symmetrical. However, the pair (1,2) is in R, but the pair (2,1) is not in R. Therefore, the relation R is not symmetric.

Transitive Property:
A relation R on a set A is transitive if for every pair (a,b) and (b,c) in R, the pair (a,c) is also in R.

In this case, the relation R={(2,2),(3,3),(4,4),(1,2)} only contains the pairs (2,2), (3,3), and (4,4), which are not sufficient to check the transitive property. However, we can see that (2,2) and (1,2) are in R, and (2,2) and (1,2) together imply that (1,2) is related to itself. This implies that (1,2) and (1,2) should also be in R for the relation to be transitive. Since (1,2) is already in R, the relation R is transitive.

Conclusion:

Based on the above analysis, we can conclude that the given relation R={(2,2),(3,3),(4,4),(1,2)} is not reflexive, not symmetric, but it is transitive. Therefore, the correct answer is option 'C' - transitive.

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