To analyze the relation $R = \{(2, 4), (4, 2)\}$ on the set $A = \{1, 2, 3, 4\}$, let's examine its properties: reflexivity, irreflexivity, antisymmetry, symmetry, and transitivity.

1. Reflexivity:

A relation is reflexive if for every element $a \in A$, the pair $(a, a)$ is in $R$.

• Since $R$ does not contain any pairs like $(1, 1), (2, 2), (3, 3),$ or $(4, 4)$, the relation is not reflexive.

2. Irreflexivity:

A relation is irreflexive if for every element $a \in A$, the pair $(a, a)$ is not in $R$.

• As no pair of the form $(a, a)$ exists in $R$, the relation is irreflexive.

3. Antisymmetry:

A relation is antisymmetric if for all $a, b \in A$, whenever $(a, b) \in R$ and $(b, a) \in R$, it must be that $a = b$.

• Here, $(2, 4)$ and $(4, 2)$ are both in $R$, but $2 \neq 4$. Therefore, the relation is not antisymmetric.

4. Symmetry:

A relation is symmetric if for every $(a, b) \in R$, $(b, a)$ is also in $R$.

• Since $(2, 4)$ and $(4, 2)$ are both in $R$, the relation is symmetric.

5. Transitivity:

A relation is transitive if whenever $(a, b) \in R$ and $(b, c) \in R$, then $(a, c)$ must also be in $R$.

• For $(2, 4)$ and $(4, 2)$, there is no $(2, 2)$ in $R$, nor does any other pair satisfy this condition. Thus, the relation is not transitive.

Conclusion:

Based on the analysis:

• The relation is irreflexive (no $(a, a)$ pairs in $R$).
• The relation is not antisymmetric (since $(2, 4)$ and $(4, 2)$ are in $R$).
• The relation is not transitive.
• The relation is symmetric.

The correct options are:

• Irreflexive, Not antisymmetric and Not transitive
• Not reflexive, Not antisymmetric and Not transitive

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Would you like further details or have any questions? Here are some related questions:

1. How is antisymmetry different from symmetry in relations?
2. What are some examples of irreflexive relations in everyday life?
3. How can we determine if a relation is an equivalence relation?
4. What are some practical applications of transitive relations?
5. Can a relation be both symmetric and antisymmetric? Why or why not?
6. How does reflexivity relate to the concept of identity in mathematics?
7. What happens if a relation is both reflexive and transitive?
8. Can a non-reflexive relation be transitive?

Tip: When determining the properties of a relation, systematically check each property (reflexivity, symmetry, transitivity, etc.) to ensure a complete understanding.