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Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack.
a) {(a, b) | a and b are the same age}
b) {(a, b) | a and b have the same parents}
c) {(a, b) | a and b share a common parent}
d) {(a, b) | a and b have met} e) {(a, b) | a and b speak a common language}

Respuesta :

varshamittal029

Relations in the options (a), (b) and (e) are the Equivalence relations as, the relations are reflexive, symmetric and transitive all.

Given, five relations as,

a) {(a, b) | a and b are the same age}

b) {(a, b) | a and b have the same parents}

c) {(a, b) | a and b share a common parent}

d) {(a, b) | a and b have met}

e) {(a, b) | a and b speak a common language}

we have to find which of them are equivalence relations

a) as, (a , a) ∈ R ∀ a hence, Reflexive.

also, if (a , b) ∈ R then (b , a) ∈ R ∀ a, b. Hence, Symmetric.

and if (a , b) ∈ R & (b , c) ∈ R then (a , c) ∈ R ∀ a, b, c. Hence, Transitive.

So, R = {(a, b) | a and b are the same age} is an equivalence relation.

b) as, (a , a) ∈ R ∀ a hence, Reflexive.

also, if (a , b) ∈ R then (b , a) ∈ R ∀ a, b. Hence, Symmetric.

and if (a , b) ∈ R & (b , c) ∈ R then (a , c) ∈ R ∀ a, b, c. Hence, Transitive.

So, R = {(a, b) | a and b have the same parents} is an equivalence relation.

c) as, (a , a) ∈ R ∀ a hence, Reflexive.

also, if (a , b) ∈ R then (b , a) ∈ R ∀ a, b. Hence, Symmetric.

and if (a , b) ∈ R & (b , c) ∈ R then (a , c) need not to be in R for some a, b, c. Hence, Non - Transitive.

So, R = {(a, b) | a and b share a common parent} is not an equivalence relation.

d) as, (a , a) ∈ R ∀ a hence, Reflexive.

also, if (a , b) ∈ R then (b , a) ∈ R ∀ a, b. Hence, Symmetric.

and if (a , b) ∈ R & (b , c) ∈ R then (a , c) need not to be in R for some a, b, c. Hence, Non - Transitive.

So, R = {(a, b) | a and b have met} is not an equivalence relation.

e) as, (a , a) ∈ R ∀ a hence, Reflexive.

also, if (a , b) ∈ R then (b , a) ∈ R ∀ a, b. Hence, Symmetric.

and if (a , b) ∈ R & (b , c) ∈ R then (a , c) ∈ R ∀ a, b, c. Hence, Transitive.

So, R = {(a, b) | a and b speak a common language} is an equivalence relation.

Hence, the relations in options (a) , (b) and (e) are the equivalence relations.

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