Question

Let Z be the set of integers. Show that the relation R = { (a,b) : a,b $\in$ Z and a + b is even }
is an equivalence relation on Z.

Answer 1

Hint: First we are going to look at the definition of symmetric, reflexive and transitive. And after that we will check all the three conditions and if it satisfies the condition for symmetric, reflexive and transitive then we can say that it is on equivalence relation on Z.

Complete step-by-step answer:
Let’s start our solution by first writing all the definition of the terms:
Symmetric: If we have a set containing two elements ‘a’ and ‘b’, then if the relation set has (a,b) then it must have (b,a) then we can say that it is symmetric.
Reflexive: If we have a set containing two elements ‘a’ and ‘b’, then if the relation set has (a,a) and (b,b) then we can say it is reflexive.
Transitive: If we have a set containing three elements ‘a’ , ‘b’, and ‘c’ then if the relation set has (a,b) and (b,c) then it must have (a,c) for transitive.
Now we have stated all the required definitions.
Now we will check each of them,
$\left( a,a \right)\in R$ , a + a = 2a
2a is an even number.
Hence it is reflexive.
If $\left( a,b \right)\in R$ then a + b is an even number,
$\Rightarrow b+a$ is also an even number.
$\Rightarrow \left( b,a \right)\in R$
Hence it is symmetric.
If $\left( a,b \right)\in R$ and $\left( b,c \right)\in R$ then a + b is an even number and b + c is an even number.
Now if we add any two even numbers then the result will also be an even number.
Therefore,
$\left( a+b \right)+\left( b+c \right)$ is also an even number.
$\left( a+c \right)+2b$ is an even number.
Now we know that 2b is even,
So, a + c can also be an even number.
$\left( a,c \right)\in R$
Hence, it is also transitive.
The relation is called equivalence when it is reflexive, symmetric and transitive.
Hence, R is an equivalence relation on Z.

Note: The definition of the terms that we have used is very important, without understanding it’s meaning clearly we cannot solve this question. The important part is to understand how we have proved transitive by adding two even numbers.