Answer:
R is an equivalence relation, since R is reflexive, symmetric, and transitive.
Step-by-step explanation:
The relation R is an equivalence if it is reflexive, symmetric and transitive.
The order to options required to show that R is an equivalence relation are;
((a, b), (a, b)) ∈ R since a·b = b·a
Therefore, R is reflexive
If ((a, b), (c, d)) ∈ R then a·d = b·c, which gives c·b = d·a, then ((c, d), (a, b)) ∈ R
Therefore, R is symmetric
If ((c, d), (e, f)) ∈ R, and ((a, b), (c, d)) ∈ R therefore, c·f = d·e, and a·d = b·c
Multiplying gives, a·f·c·d = b·e·c·d, which gives, a·f = b·e, then ((a, b), (e, f)) ∈R
Therefore R is transitive
From the above proofs, the relation R is reflexive, symmetric, and transitive, therefore, R is an equivalent relation.
Reasons:
Prove that the relation R is reflexive
Reflexive property is a property is the property that a number has a value that it posses (it is equal to itself)
The given relation is ((a, b), (c, d)) ∈ R if and only if a·d = b·c
By multiplication property of equality; a·b = b·a
Therefore;
((a, b), (a, b)) ∈ R
The relation, R, is reflexive.
Prove that the relation, R, is symmetric
Given that if ((a, b), (c, d)) ∈ R then we have, a·d = b·c
Therefore, c·b = d·a implies ((c, d), (a, b)) ∈ R
((a, b), (c, d)) and ((c, d), (a, b)) are symmetric.
Therefore, the relation, R, is symmetric.
Prove that R is transitive
Symbolically, transitive property is as follows; If x = y, and y = z, then x = z
From the given relation, ((a, b), (c, d)) ∈ R, then a·d = b·c
Therefore, ((c, d), (e, f)) ∈ R, then c·f = d·e
By multiplication, a·d × c·f = b·c × d·e
a·d·c·f = b·c·d·e
Therefore;
a·f·c·d = b·e·c·d
a·f = b·e
Which gives;
((a, b), (e, f)) ∈ R, therefore, the relation, R, is transitive.
Therefore;
R is an equivalence relation, since R is reflexive, symmetric, and transitive.
Based on a similar question posted online, it is required to rank the given options in the order to show that R is an equivalence relation.
Learn more about equivalent relations here:
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