Let R be a relation on N*N such that R={(a,b),(c,d)} : ad(b+c)=bc(a+d)} then R is
1. Reflexive 2. Symmetric

3.Transitive 4. Equivalenence

To determine the properties of the relation R defined on the set of natural numbers N*N, we need to analyze the relation given by the condition \( ad(b+c) = bc(a+d) \). Let's break this down step by step to see if R is reflexive, symmetric, transitive, and ultimately an equivalence relation.

Reflexivity

A relation is reflexive if every element is related to itself. In our case, we need to check if for every \( (a, b) \in R \), the condition holds when we substitute \( (a, b) \) into the relation.

Substituting \( (a, b) \) into the equation gives us:

\( ad(b+b) = bc(a+b) \) or \( ad(2b) = bc(a+b) \).

This simplifies to checking if \( 2ab = ab(a+b) \). For this to hold for all \( a, b \), we can see that it does not necessarily hold unless \( a = 0 \) or \( b = 0 \), which are not in the set of natural numbers. Thus, R is not reflexive.

Symmetry

A relation is symmetric if whenever \( (a, b) \) is in R, then \( (b, a) \) is also in R. We need to check if the condition holds when we swap \( a \) and \( b \).

Substituting \( (b, a) \) gives us:

\( ba(c+d) = ac(b+d) \).

By rearranging, we can see that this does not necessarily hold true for all \( a, b, c, d \). Therefore, R is not symmetric.

Transitivity

A relation is transitive if whenever \( (a, b) \) and \( (b, c) \) are in R, then \( (a, c) \) must also be in R. To check this, we would need to find pairs \( (a, b) \) and \( (b, c) \) that satisfy the relation and see if \( (a, c) \) does as well.

However, since we already established that R is not reflexive or symmetric, we can conclude that it is unlikely to be transitive either. Without specific pairs to test, we can infer that the failure of the previous properties suggests that R is not transitive.

Equivalence Relation

For R to be an equivalence relation, it must be reflexive, symmetric, and transitive. Since we have established that R fails to meet the criteria for reflexivity, symmetry, and transitivity, it follows that R cannot be classified as an equivalence relation.

Summary

• Reflexive: No
• Symmetric: No
• Transitive: No
• Equivalence Relation: No

In conclusion, the relation R defined by the condition \( ad(b+c) = bc(a+d) \) does not satisfy the necessary properties to be considered reflexive, symmetric, transitive, or an equivalence relation. Each of these properties must hold for R to be classified as an equivalence relation, and since it fails all three, we can definitively state that R is not an equivalence relation.