Question

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:
\[\begin{align}
  & \left( a,b \right)R\left( c,d \right)\Leftrightarrow ad=bc~for\text{ }all~\left( a,b \right),\left( c,d \right)\in Z\times {{Z}_{0}} \\
 & Prove\text{ }that~R~is\text{ }an\text{ }equivalence\text{ }relation\text{ }on~Z\times {{Z}_{0}}. \\
\end{align}\]

Answer 1

Hint:-For solving these questions, we would be requiring knowledge about symmetric, reflexive and transitive functions.

Complete step-by-step answer:
A relation is a relationship between sets of values. In math, the relation is between the x-values and y-values of ordered pairs. The set of all x-values is called the domain, and the set of all y-values is called the range.
The types of relations are as follows
Reflexive Relation
A relation is a reflexive relation if every element of set A maps to itself. i.e. for every a \[\in \] A, (a, a) \[\in \] R.
Symmetric Relation
A symmetric relation is a relation R on a set A if (a, b) \[\in \] R then (b, a) \[\in \] R, for all a and b \[\in \] A.
Transitive Relation
If (a, b) \[\in \] R, (b, c) \[\in \] R, then (a, c) \[\in \] R, for all a, b, c \[\in \] A and this relation in set A is transitive.
Equivalence Relation
If and only if a relation is reflexive, symmetric and transitive, it is called an equivalence relation.
As mentioned in the question, we have to prove that R is an equivalence relation on \[Z\times {{Z}_{0}}\] .
Now, we can write the following properties of R as follows
 For checking the reflexivity, we can write the following
Let (a, b) \[\in Z\times {{Z}_{0}}\]
⇒ (a, b) \[\in Z\times {{Z}_{0}}\]
⇒ ab=ba
⇒ (a, b) \[\in R\] for all (a, b) \[\in Z\times {{Z}_{0}}\]
Hence, we can say that R is reflexive on \[Z\times {{Z}_{0}}\] .

For checking the symmetry, we can write the following
Let (a, b), (c, d) \[\in Z\times {{Z}_{0}}\] 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) \[\in Z\times {{Z}_{0}}\] .
Hence, we can say that R is symmetric on \[Z\times {{Z}_{0}}\] ​.
 For checking the transitivity, we can write the following
Let (a, b), (c, d), (e, f) \[\in Z\times {{Z}_{0}}\] ​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) \[\in Z\times {{Z}_{0}}\]
Therefore, we can say that R is transitive on \[Z\times {{Z}_{0}}\] .
Hence, the given relation is an equivalence relation.

Note:-The students can make an error if they don’t know about the definitions and the meaning of different types of relations.
Also, it is important to know about the different types of set representations that are the rooster form or the set builder form as without knowing these one could never understand the question properly.