Respuesta :
Answer:
The given relation R is equivalence relation.
Step-by-step explanation:
Given that:
[tex]((a, b), (c, d))\in R[/tex]
Where [tex]R[/tex] is the relation on the set of ordered pairs of positive integers.
To prove, a relation R to be equivalence relation we need to prove that the relation is reflexive, symmetric and transitive.
1. First of all, let us check reflexive property:
Reflexive property means:
[tex]\forall a \in A \Rightarrow (a,a) \in R[/tex]
Here we need to prove:
[tex]\forall (a, b) \in A \Rightarrow ((a,b), (a,b)) \in R[/tex]
As per the given relation:
[tex]((a,b), (a,b) ) \Rightarrow ab =ab[/tex] which is true.
[tex]\therefore[/tex] R is reflexive.
2. Now, let us check symmetric property:
Symmetric property means:
[tex]\forall \{a,b\} \in A\ if\ (a,b) \in R \Rightarrow (b,a) \in R[/tex]
Here we need to prove:
[tex]\forall {(a, b),(c,d)} \in A \ if\ ((a,b),(c,d)) \in R \Rightarrow ((c,d),(a,b)) \in R[/tex]
As per the given relation:
[tex]((a,b),(c,d)) \in R[/tex] means [tex]ad = bc[/tex]
[tex]((c,d),(a,b)) \in R[/tex] means [tex]cb = da\ or\ ad =bc[/tex]
Hence true.
[tex]\therefore[/tex] R is symmetric.
3. R to be transitive, we need to prove:
[tex]if ((a,b),(c,d)),((c,d),(e,f)) \in R \Rightarrow ((a,b),(e,f)) \in R[/tex]
[tex]((a,b),(c,d)) \in R[/tex] means [tex]ad = cb[/tex].... (1)
[tex]((c,d), (e,f)) \in R[/tex] means [tex]fc = ed[/tex] ...... (2)
To prove:
To be [tex]((a,b), (e,f)) \in R[/tex] we need to prove: [tex]fa = be[/tex]
Multiply (1) with (2):
[tex]adcf = bcde\\\Rightarrow fa = be[/tex]
So, R is transitive as well.
Hence proved that R is an equivalence relation.
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:
https://brainly.com/question/1503196
