To determine whether the relation $R = \{(1,1), (1,2), (2,2), (2,3), (3,3)\}$ is an equivalence relation on the set $A = \{1, 2, 3\}$, we need to check if $R$ satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity.

1. Reflexivity:

A relation $R$ on a set $A$ is reflexive if every element in $A$ is related to itself, i.e., for all $a \in A$, $(a, a) \in R$.

For $A = \{1, 2, 3\}$, we need the pairs $(1, 1), (2, 2), (3, 3)$ to be in $R$.

• $(1, 1) \in R$, which is true.
• $(2, 2) \in R$, which is true.
• $(3, 3) \in R$, which is true.

Since all elements of $A$ are related to themselves, the relation $R$ is reflexive.

2. Symmetry:

A relation $R$ on a set $A$ is symmetric if for all $a, b \in A$, whenever $(a, b) \in R$, it must also hold that $(b, a) \in R$.

Check the pairs in $R$:

• $(1, 1)$ is symmetric (since $(1, 1) = (1, 1)$).
• $(1, 2) \in R$, but $(2, 1) \notin R$, so symmetry is violated.
• $(2, 2)$ is symmetric.
• $(2, 3) \in R$, but $(3, 2) \notin R$, so symmetry is violated.
• $(3, 3)$ is symmetric.

Since symmetry is violated (for $(1, 2)$ and $(2, 3)$), the relation $R$ is not symmetric.

3. Transitivity:

A relation $R$ on a set $A$ is transitive if for all $a, b, c \in A$, whenever $(a, b) \in R$ and $(b, c) \in R$, it must also hold that $(a, c) \in R$.

Check the transitive property for the pairs in $R$:

• $(1, 1) \in R$ and $(1, 2) \in R$. We need $(1, 2) \in R$, which is true.
• $(1, 2) \in R$ and $(2, 3) \in R$. We need $(1, 3) \in R$, but $(1, 3) \notin R$, so transitivity is violated.
• $(2, 2) \in R$ and $(2, 3) \in R$. We need $(2, 3) \in R$, which is true.
• $(3, 3) \in R$, so no transitivity check is needed here.

Since the transitivity condition is violated (for $(1, 2)$ and $(2, 3)$), the relation $R$ is not transitive.

Conclusion:

Since the relation $R$ is reflexive but neither symmetric nor transitive, it is not an equivalence relation.

Would you like more details on any of the properties or steps?

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Here are 5 questions to consider based on this topic:

1. What conditions must a relation satisfy to be transitive?
2. How does the lack of symmetry impact a relation in practical scenarios?
3. Can a relation be reflexive but still not an equivalence relation? Why or why not?
4. What changes could be made to the relation $R$ to make it symmetric?
5. How is the concept of equivalence relations applied in real-world mathematical problems?

Tip: When checking if a relation is symmetric, always verify that if $(a, b)$ is in the relation, then $(b, a)$ must also be in the relation.