Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z×Z0 be defined as follows:(a,b)R(c,d)⇔ad=bc for all (a,b),(c,d)∈Z×Z0Prove that R is an equivalence relation on Z×Z0.

Question

Let Z be the set of all integers and Z0​ be the set of all non-zero integers. Let a relation R on Z×Z0​ be defined as follows:(a,b)R(c,d)⇔ad=bc for all (a,b),(c,d)∈Z×Z0​Prove that R is an equivalence relation on Z×Z0​.

Filo · Accepted Answer

We observe the following properties of RReflexivity:Let (a,b)∈Z×Z0​⇒a,b∈Z,Z0​⇒ab=ba⇒(a,b)∈R for all (a,b)∈Z×Z0​So,R is reflexive on Z×Z0​Symmetry:Let (a,b),(c,d)∈Z×Z0​ such that (a,b)R(c,d)⇒ad=bc⇒cb=da⇒(c,d)R(a,b)Thus, (a,b)R(c,d)⇒(c,d)R(a,b) for all (a,b),(c,d)∈Z×Z0​So,R is symmetric on Z×Z0​Transitivity:Let (a,b),(c,d),(e,f)∈N×N0​ such that (a,b)R(c,d) and (c,d)R(e,f)Then,(a,b)R(c,d)⇒ad=bc(c,d)R(e,f)⇒cf=de⇒(ad)(cf)=(bc)(de)⇒af=be⇒(a,b)R(e,f) for all values of (a,b),(c,d),(e,f)∈N×N0​So, R is transitive on N×N0​Hence the given relation is an equivalence relation.

Question Text	Let $Z$ be the set of all integers and $Z_{0}$ be the set of all non-zero integers. Let a relation $R$ on $Z \times Z_{0}$ be defined as follows: $(a, b) R (c, d) \Leftrightarrow a d = b c$ for all $(a, b), (c, d) \in Z \times Z_{0}$ Prove that $R$ is an equivalence relation on $Z \times Z_{0}$ .
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Let $Z$ be the set of all integers and $Z_{0}$ be the set of all non-zero integers. Let a relation $R$ on $Z \times Z_{0}$ be defined as follows:
$(a, b) R (c, d) \Leftrightarrow a d = b c$ for all $(a, b), (c, d) \in Z \times Z_{0}$
Prove that $R$ is an equivalence relation on $Z \times Z_{0}$ .

Text solutionVerified

Practice questions from similar books

Let Z be the set of all integers and Z0​ be the set of all non-zero integers. Let a relation R on Z×Z0​ be defined as follows:(a,b)R(c,d)⇔ad=bc for all (a,b),(c,d)∈Z×Z0​Prove that R is an equivalence relation on Z×Z0​.

Text solutionVerified

Practice questions from similar books

Let $Z$ be the set of all integers and $Z_{0}$ be the set of all non-zero integers. Let a relation $R$ on $Z \times Z_{0}$ be defined as follows:
$(a, b) R (c, d) \Leftrightarrow a d = b c$ for all $(a, b), (c, d) \in Z \times Z_{0}$
Prove that $R$ is an equivalence relation on $Z \times Z_{0}$ .