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Reflexivity is a fundamental property of an equivalence relation. It means that every element is related to itself. In simpler terms, for an equivalence relation \(R\) on a set \(A\), every element \(x\) in \(A\) must satisfy \((x, x) \in R\). This property ensures that each element is included in the relation in a self-referential way.

The identity relation \(I_A = \{(1,1), (2,2), (3,3), (4,4), (5,5)\}\) is an excellent example of reflexivity as it contains every element from the set \(A\) paired with itself. Any attempt to construct a new equivalence relation must ensure reflexivity is preserved. For the relation \(R = \{(1,1), (2,2), (3,3), (4,4), (5,5), (1,2), (2,1)\}\), reflexivity is naturally maintained because all the self-pairs \((x,x)\) are still present in \(R\).

This property is crucial because it lays the foundation for symmetry and transitivity. Without reflexivity, the set wouldn't be equivalently connected within itself.

Symmetry in an equivalence relation means that if one element is related to another, then the second element is also related back to the first. Formally, a relation \(R\) on set \(A\) is symmetric if for all \(x, y \in A\), whenever \((x, y) \in R\), then \((y, x) \in R\) must also hold true. This two-way relationship allows the elements to freely swap positions without breaking the connection between them.

Looking at the relation \(R = \{(1,1), (2,2), (3,3), (4,4), (5,5), (1,2), (2,1)\}\), symmetry is evident. The pair \((1,2)\) is mirrored by \((2,1)\), showing that the relation between 1 and 2 is mutual and reciprocal. It's crucial to verify that for every added pair, its symmetric counterpart is also included to maintain this essential property.

Symmetry supports balanced exchanges within a relation, creating an aura of equality and mutual connection among elements of the set \(A\). Without symmetry, equivalence relations would lack one of their defining characteristics, akin to having a lopsided connection that isn't fair or mutual.

Transitivity is the property that allows an equivalence relation to extend beyond direct pairs. If an element is related to a second, and the second is related to a third, transitivity dictates that the first element is also related to the third. Formally, for a relation \(R\) on set \(A\), it must hold that for all \(x, y, z \in A\), if \((x, y) \in R\) and \((y, z) \in R\), then \((x, z) \in R\) must be part of \(R\) as well.

In the constructed relation \(R = \{(1,1), (2,2), (3,3), (4,4), (5,5), (1,2), (2,1)\}\), transitivity can be a bit less obvious as there are fewer chains to verify. For example, look at the pairs \((1,2)\) and \((2,1)\). Although they circle back to themselves via reflexivity, this doesn't give rise to new pairs that need to be checked. This simplicity in design ensures that there are no transitive incompletions, confirming that the existing relation is closed under transitivity.

Transitivity adds a depth of connection, allowing an equivalence relation to relate not just pairs of elements but extendable sequences. This property stitches the fabric of the set into a cohesive and comprehensive equivalence relation.