Question

Determine which of the following five relations defined on \(\mathbb{Z}\) are equivalence relations: (a) \(\\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a>0\) and \(b>0)\) or \((a<0\) and \(b<0)\\}\) (b) \(\\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a \geq 0\) and \(b>0\) ) or \((a<0\) and \(b \leq 0)\\}\) (c) \(\\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:|a-b| \leq 10\\}\) (d) \(\\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a \leq 0\) and \(b \geq 0)\) or \((a \leq 0\) and \(b \leq 0)\\}\) (e) \(\\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a \geq 0\) and \(b \geq 0\) ) or \((a \leq 0\) and \(b \leq 0)\\}\)

Short answer

Only relation (e) is an equivalence relation.

Step-by-step solution

Definition of an Equivalence Relation

An equivalence relation on a set \(S\) is a relation that is reflexive, symmetric, and transitive. We will verify these properties for each relation.

Evaluate Relation (a)

The relation is defined as \(\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a>0\) and \(b>0)\) or \((a<0\) and \(b<0)\}\).- **Reflexivity**: For any \(a \in \mathbb{Z}\), \((a, a)\) must satisfy the condition. However, \(a = 0\) does not satisfy the condition, so it is not reflexive.- **Symmetry**: If \((a, b)\) is in the relation, then \((b, a)\) must also be. This is true because the same sign condition on \(a, b\) preserves symmetry.- **Transitivity**: If \((a, b)\) and \((b, c)\) are in the relation, then \((a, c)\) must also be. This condition is satisfied assuming \(a, b, c\) are of the same sign.This relation is not reflexive, hence not an equivalence relation.

Evaluate Relation (b)

The relation is \(\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a \geq 0\) and \(b>0)\) or \((a<0\) and \(b \leq 0)\}\).- **Reflexivity**: \((a, a)\) needs to hold. For \(a=0\), this doesn't satisfy; hence not reflexive.- **Symmetry**: There exist pair \((a, b)\) where \((b, a)\) will not satisfy; hence not symmetric.- **Transitivity**: Does not hold for all cases.This relation is neither reflexive, symmetric, nor transitive, hence not an equivalence relation.

Evaluate Relation (c)

The relation is \(\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:|a-b| \leq 10\}\).- **Reflexivity**: For any \(a \in \mathbb{Z}\), \(|a-a| = 0 \leq 10\), so the relation is reflexive.- **Symmetry**: If \(a\) is within 10 units of \(b\), then \(b\) is within 10 units of \(a\), thus symmetric.- **Transitivity**: If \(|a-b| \leq 10\) and \(|b-c| \leq 10\), then \(|a-c| \leq |a-b| + |b-c| \leq 20\), but for a smaller threshold like 10, it fails; hence not transitive.This relation fails transitivity, hence not an equivalence relation.

Evaluate Relation (d)

The relation is \(\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a \leq 0\) and \(b \geq 0)\) or \((a \leq 0\) and \(b \leq 0)\}\).- **Reflexivity**: \((a, a)\) does not satisfy as both conditions contradict; hence not reflexive.- **Symmetry**: Cannot be symmetric as \((a, b)\) does not imply \((b, a)\).- **Transitivity**: Also not transitive.This relation lacks all required properties for an equivalence relation.

Evaluate Relation (e)

The relation is \(\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a \geq 0\) and \(b \geq 0)\) or \((a \leq 0\) and \(b \leq 0)\}\).- **Reflexivity**: \((a, a)\) for any \(a\) holds as both sides are either non-negative or non-positive.- **Symmetry**: If both belong to the same category (non-negative or non-positive), it's symmetrical.- **Transitivity**: If \((a, b)\) and \((b, c)\), both will belong to the same category, hence transitive.This relation satisfies reflexivity, symmetry, and transitivity, hence it is an equivalence relation.

Key concepts

Reflexivity

When we talk about the reflexivity of a relation, it means that every element relates to itself. Let's imagine we have a set of numbers. To say a relation is reflexive on this set, each number in the set must pair with itself. For example, in set notation, if the set is denoted as \( S \), the pairs \((a, a)\) must be included for every \( a \) in \( S \).
Why is reflexivity important? Well, it acts like a self-check. It ensures that no element is left out or treated differently in the relational aspect of equivalence. Let's use the example from the step-by-step solution and analyze relation (e): In this case, \((a, a)\) will always meet the condition if \( a \) is either non-negative or non-positive. This ensures that the reflexive property holds for relation (e), making it part of an equivalence relation.
However, in relation (a), reflexivity fails because, for example, zero cannot relate to itself under the given condition. Reflexivity is a crucial test for identifying equivalence relations.

Symmetry

A relation is termed symmetric if whenever one element is related to another, the second element is also related to the first. Think of symmetry as a two-way street: if you can drive from one end to the other, you can also return.

• If \((a, b)\) is in the relation, then \((b, a)\) should also be there for symmetry to hold.
• This means both sides of the pair are on an equal footing.

To give you an example, consider relation (e) in the original solution. Here, if \(a\) and \(b\) are both non-negative or both non-positive, then symmetry holds because switching \(a\) and \(b\) doesn't affect the conditions being met.
On the other hand, relation (b) doesn't satisfy the symmetry condition, as pointed out in the solution. This lack of symmetry occurs when an exchange of \(a\) and \(b\) doesn't meet the original criteria set by the relation.
In essence, if symmetry fails in a relation, it cannot be an equivalence relation. Symmetry ensures fairness and consistency, integral components for equivalency.

Transitivity

Transitivity connects pairs through a common element, establishing a continuity chain. It's like making friends through mutual acquaintances! In formal terms, if a relation is transitive, whenever you have \((a, b)\) and \((b, c)\) in your relation set, then \((a, c)\) should also be in the set.

• This means you can hop from one element to another through related elements.
• It's a form of *logical sequence* that extends equivalence from direct pairs to indirect pairs.

In the provided solution, relation (e) is transitive because if both pairs \((a, b)\) and \((b, c)\) are satisfied under either both being non-negative or both non-positive, \((a, c)\) naturally follows the same pattern.
However, take a look at relation (c), where this fails. Since the relation given by \(|a-b| \leq 10\) and \(|b-c| \leq 10\) doesn't guarantee that \(|a-c| \leq 10\), transitivity collapses. It forgets to consider cumulative effect or how conditions link across sequences of elements.
Transitivity ensures the *domino effect* in any sequence of relations and is indispensable for a relation to be an equivalence relation. Without it, the "chain" would break, preventing the establishment of universal equivalence.