Equivalence Relations

The relation in mathematics would be the relationship between two or more sets of values.

An equivalence relation is really a type of binary relation in mathematics that should be reflexive, symmetric, and transitive. The “equal to (=)” relation is a very well example of an equivalence relation. That is, whenever two elements of the given set belong to the same equivalence class, they are equivalent.

Such that if a binary relation ∼ on a set A is reflexive, symmetric, and transitive, it is said to be an equivalence relation.

The examples given can help to explain equivalence relations:

• On a set of numbers, the sign of ‘is equal to (=)’; for example, 1/3 = 3/9.
• The relationship of ‘is similar to (~)’ and ‘is congruent to (≅)’ for a given set of triangles demonstrates equivalence.
• The relation ‘congruence modulo n (≡)’ demonstrates equivalence for a given set of integers.
• The image and domain are the same under a function, demonstrating the equivalence relation.
• ‘has the same cosine’ for a set of all angles.
• ‘has the same absolute value’ for a set of all real numbers.

Equivalence Relation Proof

Here’s an example of an equivalence relation to demonstrating the properties.

Assume R is a relation on the set of ordered pairs of positive integers that ((a, b), (c, d)) ∈ R if and only if ad=bc. Can R be an equivalence relation?

To demonstrate that R is an equivalence relation, we must demonstrate that it is reflexive, symmetric, and transitive.

The following is the proof for the given condition:

Reflexive Property

As per the reflexive property, if (a, a) ∈ R, for every a∈A

For every pair of positive integers,

((a, b),(a, b))∈ R.

Easily, we can say

ab = ab for all positive integers.

Thus, the reflexive property is proved.

Symmetric Property

As per the symmetric property, if (a, b) ∈ R, then we can say (b, a) ∈ R

According to a given condition,

if ((a, b),(c, d)) ∈ R, then ((c, d),(a, b)) ∈ R.

If ((a, b),(c, d))∈ R, then ad = bc and cb = da

We know that multiplication is commutative.

Thus, ((c, d),(a, b)) ∈ R

Therefore, the symmetric property is proved.

Transitive Property

As per the transitive property, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) also belongs to R

We have the set of ordered pairs of positive integers,

((a, b), (c, d))∈ R and ((c, d), (e, f))∈ R,

then ((a, b),(e, f) ∈ R.

It can be assume that ((a, b), (c, d))∈ R and ((c, d), (e, f)) ∈ R.

So, we get, ad = cb and cf = de.

Such a relation implies that a/b = c/d and that c/d = e/f,

Then, a/b = e/f we get af = be.

Thus, ((a, b),(e, f))∈ R.

Therefore, the transitive property is proved.

FAQs

Q. How do you determine equivalence relations?

Ans: An equivalence relation can be defined as a relation R on a set A that is reflexive, symmetric, and transitive. The equivalence relation is really a set relationship that is commonly represented by the symbol “∼”

Q. What are the applications of equivalence relations in real life?

Ans: Once we decide that two objects are “essentially the same” according to some criterion, we establish an equivalence relation. Colour is indeed a common example from everyday life: we say two objects are equivalent if they have the same colour.

For more visit Equal and Equivalent Sets – Definition, Explanation, Examples and FAQs