Question

$N$ is the set of positive integers. The relation $R$ is defined on N x N as follows: $(a,b)R(c,d)⟺ad=bc$ Prove that

• A. $R$ is an equivalence relation.
• B. $R$ is symmetric relation.
• C. $R$ is transitive relation.
• D. $R$ is not an equivalence relation.

Answer 1

Answer: (A), (B), (C)

$R$ is an equivalence relation.
$R$ is symmetric relation.
$R$ is transitive relation.

Reflexive: let $(a,b)$ be an ordered pair of positive integer. to show R is reflexive we must show $((a,b),(a,b))\in R$. Multiplication of integer is commutative, so $ab=ba$. Thus $((a,b),(a,b))\in R$
Symmetric: Let $(a,b)$ and $(c,d)$ be an ordered pair of positive integer such that $(a,b)R(c,d)$. Then $ad=bc$. This equation is equivalent to $cb=da$, so $(c,d)R(a,b)$. This shows R is symmetric
Transitive: Let $(a,b),(c,d)$ and $(e,f)$ be ordered pairs of positive integer such that $(a,b)R(c,d)$ and $(c,d)R(e,f)$. Then $ad=bc$ and $cf=de$.
Thus $adf=bcf$ and $bcf=bde$, which implies $adf=bde$.
Since $d\ne 0$, we can cancel it from both sides of this equation to get $af=be$.
This shows $(a,b)R(e,f)$ and R is transitive