Analyzing Relation R Properties on Set I

The question asks us to identify which properties the given relation R does not possess on the set I. The relation is defined as R = {(4,5), (5,4), (7,6), (6,7)} and the set is I = {4, 5, 6, 7}. We need to check for the properties: Reflexive, Symmetric, Transitive, and Antisymmetric.

Checking for Reflexive Property

A relation R on a set I is called reflexive if for every element '$a$' in I, the pair '$(a, a)$' is also in R.

For the set I = {4, 5, 6, 7}, a reflexive relation must contain the following pairs:

• (4, 4)
• (5, 5)
• (6, 6)
• (7, 7)

Looking at the given relation R = {(4,5), (5,4), (7,6), (6,7)}, none of these pairs $(a, a)$ are present. Therefore, the relation R is not reflexive.

Checking for Symmetric Property

A relation R on a set I is called symmetric if whenever a pair '$(a, b)$' is in R, the pair '$(b, a)$' must also be in R.

Let's examine the pairs in R:

• For the pair (4,5) in R, the pair (5,4) is also present in R.
• For the pair (5,4) in R, the pair (4,5) is also present in R.
• For the pair (7,6) in R, the pair (6,7) is also present in R.
• For the pair (6,7) in R, the pair (7,6) is also present in R.

Since for every pair $(a, b)$ in R, the pair $(b, a)$ is also in R, the relation R is symmetric.

Checking for Transitive Property

A relation R on a set I is called transitive if for any elements '$a$', '$b$', and '$c$' in I, whenever both '$(a, b)$' and '$(b, c)$' are in R, then the pair '$(a, c)$' must also be in R.

Let's check the conditions for transitivity:

• We have $(4,5) \in R$ and $(5,4) \in R$. For transitivity, $(4,4)$ must be in R. However, $(4,4) \notin R$.
• We have $(5,4) \in R$ and $(4,5) \in R$. For transitivity, $(5,5)$ must be in R. However, $(5,5) \notin R$.
• We have $(7,6) \in R$ and $(6,7) \in R$. For transitivity, $(7,7)$ must be in R. However, $(7,7) \notin R$.
• We have $(6,7) \in R$ and $(7,6) \in R$. For transitivity, $(6,6)$ must be in R. However, $(6,6) \notin R$.

Since these conditions are not met for the existing pairs, the relation R is not transitive.

Checking for Antisymmetric Property

A relation R on a set I is called antisymmetric if for any elements '$a$' and '$b$' in I, whenever both '$(a, b)$' and '$(b, a)$' are in R, it must imply that '$a$' is equal to '$b$' (i.e., $a = b$).

Let's examine the pairs where both $(a, b)$ and $(b, a)$ exist in R:

• The pairs (4,5) and (5,4) are both in R. Here, $a=4$ and $b=5$. Since $a \neq b$ ($4 \neq 5$), the condition for antisymmetry fails.
• The pairs (7,6) and (6,7) are both in R. Here, $a=7$ and $b=6$. Since $a \neq b$ ($7 \neq 6$), the condition for antisymmetry fails again.

Because we found instances where $(a, b) \in R$ and $(b, a) \in R$ but $a \neq b$, the relation R is not antisymmetric.

Conclusion on Relation Properties

Based on the analysis:

• The relation R is not reflexive.
• The relation R is symmetric.
• The relation R is not transitive.
• The relation R is not antisymmetric.

The properties that the relation R does not have are Reflexive, Transitive, and Antisymmetric. This corresponds to options A, C, and D.

Therefore, the correct choice is the one that lists A, C, and D only.