Question

Suppose \(R\) is a symmetric and transitive relation on a set \(A,\) and there is an element \(a \in A\) for which \(a R x\) for every \(x \in A .\) Prove that \(R\) is reflexive.

Short answer

After applying the symmetric property of the given relation \(R\) to the hypothesis, we find that \(x R a\) holds for all \(x \in A\). This fact along with \(a R x\) allows us to apply the transitivity of \(R\), yielding \(x R x\) for every \(x \in A\). Hence, according to the definition, \(R\) is reflexive.

Step-by-step solution

Use Symmetry

The prompt states that the relation \(R\) is symmetric, which means that for all \(x, y\) in \(A\), if \(x R y\) then \(y R x\). We are given that there exists an element \(a\) in \(A\) such that \(a R x\) holds true for every \(x\) in \(A\). Thus, using the symmetry property, it can be said that for every \(x\) in \(A\), \(x R a\).

Apply Transitivity

In addition, the relation \(R\) is also said to be transitive, from this we know that if \(x R a\) and \(a R x\), then \(x R x\). Since the first part of transitivity condition is established in Step 1 (\(x R a\) for all \(x\) in \(A\)) and the latter part (\(a R x\)) is part of the problem's statement, it follows, thanks to transitivity, that \(x R x\) for every \(x\) in \(A\).

Concluding Reflexivity

Recall the definition of a reflexive relation: a relation \(R\) is reflexive if \(x R x\) for all \(x\) in \(A\). As per the conclusion in Step 2, \(x R x\) holds true for every \(x\) in \(A\), and hence, by definition, the relation \(R\) is reflexive.

Key concepts

Symmetric Relation

A symmetric relation is a type of binary relation where, for any elements a and b in a set, if a is related to b, then b is necessarily related to a. This characteristic is formalized as: if aRb, then bRa. The concept is important for satisfying many mathematical properties and it plays a role in the definition of equivalence relations.

In the context of the given exercise, the symmetry of the relation meant that when an element a from the set was related to any other element x, it implied that x was also related back to a. Understanding the symmetric property provides clarity as to why, once we have established a relation from a to every x, it means every x is also related back to a, which subsequently helps in proving reflexivity when combined with transitivity.

Transitive Relation

A transitive relation further shapes the structure of a set with its distinct property: for any elements a, b, and c in a set, if a is related to b, and b is related to c, then a must also be related to c. In mathematical terms, if aRb and bRc, it logically follows that aRc. This concept is crucial when dealing with ordered sets and is a foundation for analyzing sequences and series.

When solving the initial problem, transitivity plays a critical role after the use of symmetry. Because we know from symmetry that for every x in the set, xRa, and we are given that aRx from the problem statement, the transitive property allows us to conclude that xRx for all elements x in the set. This step is vital in demonstrating the reflexive nature of the relation R.

Reflexivity in Mathematics

Reflexivity in mathematics refers to a relation's inclusion of every element with itself. Specifically, a reflexive relation on a set requires that every element is related to itself. For a set A and a relation R defined on it, mathematicians express this idea by stating xRx for every element x in A. This concept is a fundamental part of identity relations and is often assumed in common mathematical structures such as the real numbers with the relation of equality.

In our example where the set A and relation R are given, we used the combined properties of symmetry and transitivity to infer reflexivity. By establishing that for all elements x in A, xRx holds true, we matched the definition of a reflexive relation. This is what ultimately allowed us to conclude that relation R is indeed reflexive, based on the given premises. When learning about mathematical relations, appreciating the interconnectedness of these properties can deepen understanding and problem-solving abilities.