Question

If \(\mathrm{A}=\\{1,2,3\\}\), then the number of equivalence relation containing \((1,2)\) is (a) 1 (b) 2 (c) 3 (d) 8

Short answer

The number of equivalence relations containing (1, 2) in the set A = {1, 2, 3} is \[\boxed{(b) \ 2}\].

Step-by-step solution

List all possible relations in the set A

List all possible relations in the set A, including the trivial case where every element is related to itself: 1. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} 2. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (1, 3), (3, 2), (2, 3), (3, 3)} We will now analyze each of these relations to see if they satisfy the properties of equivalence relations.

Check for reflexivity

For a relation to be an equivalence relation, it must be reflexive. This means that (a, a) must be in the relation for every element a in the set A. Both relations have this property, so both can continue to the next step.

Check for symmetry

For a relation to be an equivalence relation, it must be symmetric, which means (a, b) is in the relation if and only if (b, a) is in the relation. We are given that (1, 2) must be included in the relation, so both relations must also include (2, 1). Looking at the relations we listed in Step 1, both relations have this property, so both can continue to the next step.

Check for transitivity

For a relation to be an equivalence relation, it must be transitive. This means that if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation. Let's check both relations: 1. Relation 1: Since (1, 2) and (2, 1) are in the relation, (1, 1) must also be in the relation (which it is). (1, 2) and (2, 3) are not in the relation, so the transitivity property is satisfied. 2. Relation 2: Since (1, 2) and (2, 3) are in the relation, (1, 3) must also be in the relation (which it is). Since (1, 3) and (3, 2) are in the relation, (1, 2) must also be in the relation (which it is). Thus, the transitivity property is satisfied. Both relations satisfy the transitivity property, so both are equivalence relations.

Count the equivalence relations containing (1, 2)

We have found 2 equivalence relations including the pair (1, 2). Thus, the answer to the problem is: (b) 2

Key concepts

Reflexivity

Reflexivity is one of the essential properties of an equivalence relation. When a relation is reflexive, every element in the set relates to itself. In more formal terms, for every element \(a\) in a set \(A\), the pair \((a, a)\) must be included in the relation. Using the set \(A = \{1, 2, 3\}\), an equivalence relation must include the pairs:

• \((1, 1)\)
• \((2, 2)\)
• \((3, 3)\)

These pairs ensure every member of the set has a direct relationship with itself. Without this property, a relation cannot be considered reflexive, and as a result, it can't be part of an equivalence relation. Considering reflexivity helps create the foundation upon which the other properties of symmetry and transitivity can comfortably rest.

Symmetry

Symmetry in relations means that if one element is related to another, the reverse must also be true. To describe this mathematically, if \((a, b)\) is in the relation, then \((b, a)\) should also be in the relation for symmetry to hold. In our given scenario within the set \(A = \{1, 2, 3\}\), we start with the given pair \((1, 2)\). Due to the requirement for symmetry, the pair \((2, 1)\) must also be included in any relation that aims to be an equivalence relation.

• This reciprocal relationship is what gives symmetry its defining feature.
• Symmetry ensures that relationships are bidirectional.

For any listed relations, if symmetry is missing, that relation can no longer qualify as an equivalence relation. Both listed relations in the exercise exhibit symmetry, meaning any such relation ensures every connection is mutual.

Transitivity

Transitivity is another crucial property of equivalence relations, which creates a chain-like connection among elements. If the relation includes \((a, b)\) and \((b, c)\), then to be transitive, \((a, c)\) must also be included in the relation. Consider the example within our set \(A = \{1, 2, 3\}\). Suppose we have both \((1, 2)\) and \((2, 3)\) in our relation. Transitivity would demand that \((1, 3)\) is also part of this relation.

• This property ensures logical consistency and the ability to "carry over" relationships through transitive links.
• Without transitivity, the chain of connectivity would break, leaving the relation incomplete.

In the problem exercise, both candidate relations respect the rule of transitivity, thereby satisfying this important feature and maintaining their status as equivalence relations.