Question

Which one of the following relations on \(R\) is an equivalence relation? (a) a \(\mathrm{R}_{1} \mathrm{~b} \Leftrightarrow|\mathrm{a}|=|\mathrm{b}|\) (b) a \(\mathrm{R}_{2} \mathrm{~b} \Leftrightarrow \mathrm{a} \geq \mathrm{b}\) (c) a \(\mathrm{R}_{3} \mathrm{~b} \Leftrightarrow\) a divides \(\mathrm{b}\) (d) a \(\mathrm{R}_{4} \mathrm{~b} \Leftrightarrow \mathrm{a}<\mathrm{b}\)

Short answer

The equivalence relation is R1, a ~ b if \(|a| = |b|\).

Step-by-step solution

Test for reflexivity

In this step, we will check if each relation is reflexive. A relation is reflexive if for every element a in the set, the relation aRa holds. (a) For the relation R1, a ~ b if |a| = |b|. Since |a| = |a| for every real number a, the relation R1 is reflexive. (b) For the relation R2, a ~ b if a ≥ b. Since a ≥ a for every real number a, the relation R2 is reflexive. (c) For the relation R3, a ~ b if a divides b. Since every real number a divides itself, the relation R3 is reflexive. (d) For the relation R4, a ~ b if a < b. Since a < a is false for every real number a, the relation R4 is not reflexive.

Test for symmetry

In this step, we will test each relation for symmetry. A relation is symmetric if for every pair of elements a and b, if a ~ b then b ~ a. (a) For R1, a ~ b if |a| = |b|. Since equality is symmetric, if |a| = |b| then |b| = |a|, so R1 is symmetric. (b) For R2, a ~ b if a ≥ b. If a ≥ b, it is not necessarily true that b ≥ a. For example, if a = 2 and b = 1, a ≥ b but b is not greater than or equal to a. Therefore, R2 is not symmetric. (c) For R3, a ~ b if a divides b. If a divides b, it does not necessarily mean that b divides a. For example, if a = 2 and b = 4, a divides b but b does not divide a. Therefore, R3 is not symmetric. (d) For R4, a ~ b if a < b. If a < b, it is not true that b < a. Therefore, R4 is not symmetric.

Test for transitivity

In this step, we will test each relation for transitivity. A relation is transitive if for every three elements a, b, and c, if a ~ b and b ~ c, then a ~ c. (a) For R1, a ~ b if |a| = |b|. If |a| = |b| and |b| = |c|, then |a| = |c| due to the transitivity of equality. Therefore, R1 is transitive. (b) For R2, a ~ b if a ≥ b. Since R2 is not symmetric, we don't need to test it for transitivity. (c) For R3, a ~ b if a divides b. Since R3 is not symmetric, we don't need to test it for transitivity. (d) For R4, a ~ b if a < b. Since R4 is not symmetric or reflexive, we don't need to test it for transitivity.

Determine the equivalence relation

Now that we have tested each relation for reflexivity, symmetry, and transitivity, we can conclude that the only relation that satisfies all three properties and is therefore an equivalence relation is R1, a ~ b if |a| = |b|.

Key concepts

Reflexivity

Reflexivity is a foundational concept in the study of equivalence relations. An equivalence relation on a set is reflexive if every element is related to itself. In other words, for any element a in the set, it must satisfy the condition aRa, where R is the relation being considered. This implies a sort of 'self-acceptance' for every member of the set.

For example, in the relation R1, where a ~ b is defined by |a| = |b|, every real number satisfies the reflexive property because the absolute value of any number is equal to itself. This makes R1 reflexive. In contrast, relation R4, defined as a < b, is not reflexive since no number is less than itself; hence it does not meet this fundamental property of equivalence relations.

Understanding reflexivity helps us eliminate potential equivalence relations that do not consider each element in relation to itself, an essential step towards fully grasping this mathematical concept.

Symmetry

Symmetry in the context of equivalence relations is a bit like perfect mirroring in everyday life. If an element a is related to an element b in a set, then b must also be related to a, denoted as aRb implies bRa. For a relation to be symmetric, it must hold true for all pairs of elements in the set.

Looking at our example, relation R1, the symmetry is evident because if the absolute value of a is equal to the absolute value of b, then the reverse is also true, satisfying the symmetric condition. Conversely, relation R2 fails the symmetry test since a greater or equal to relationship does not imply a two-way connection. The concept of symmetry is a crucial piece that ensures each pair of elements holds a bidirectional or mutual relationship within the set.

Transitivity

Transitivity takes the concept of equivalence relations one step further and ensures that the relationship 'passes through' the elements consistently. If an element a is related to b, and b is related to c, then a should also be related to c, formally expressed as aRb and bRc implies aRc. This connects elements to one another via a 'chain' of relations.

In the relation R1, transitivity is easy to observe; if |a| = |b| and |b| = |c|, it's clear that |a| must equal |c|. This doesn't hold for relations that failed earlier tests, like R2, making them unsuitable as equivalence relations. Transitivity extends the interconnectivity of elements across the whole set, which is indispensable for a relation to be an equivalence relation.