Question

Let R be a relation on the set of integers given by $aRb \Leftrightarrow a = {2^k}b$ for some integer k. Then R is
A) An equivalence relation
B) Reflexive and transitive but not symmetric
C) Reflexive and transitive but not transitive
D) Symmetric and transitive but not Reflexive

Answer 1

A relation on set A is called an equivalence relation if it is reflexive, symmetric, and transitive.
Reflexive relation: A binary relation R over a set X is reflexive if it relates every element of X to itself. Formally this may be written $\forall x \in X:xRx$
Symmetric relation: It is a type of binary relation. A binary relation R over a set X is symmetric if
$\forall a,b \in X:\left( {aRb \Leftrightarrow bRa} \right)$
Transitive relation: A homogeneous relation R on the set X is a transitive relation is, for all $a,b,c \in X$, if aRb and bRc, then aRc. Or in terms of first-order logic:
$\forall a,b,c \in X:\left( {aRb \wedge bRc} \right) \Rightarrow aRc$

Complete step-by-step solution:
First, let us check for reflexive relations.
Let us assume any integer a.
We have,
$\Rightarrow a = {2^k}a$
Take the value of k is equal to 0.
Therefore,
$\Rightarrow a = {2^0}a$
As we know that the value of ${2^0}$ is 1.
$\Rightarrow a = a$
So, we can say that
$\Rightarrow \left( {a,a} \right) \in R$
Hence R is reflexive on Z.
Now, let us check for symmetric relations.
Let $\left( {a,b} \right) \in R$
Then, for some integer k
$a = {2^k}b$
Let us divide both sides by ${2^k}$
So,
$\Rightarrow \dfrac{a}{{{2^k}}} = \dfrac{{{2^k}b}}{{{2^k}}}$
That is equal to
$b = {2^{ - k}}a$
Hence R is symmetric on Z.
Now, let us check for transitive relations.
Let $\left( {a,b} \right) \in R$ and $\left( {b,c} \right) \in R$.
$a = {2^k}b$ and $b = {2^m}c$ for some integers k and m.
$\Rightarrow a = {2^{k + m}}c$
$\Rightarrow \left( {a,c} \right) \in R$
Hence R is transitive on Z.
Therefore, R is an equivalence relation on Z.

Option A is the correct answer.

Note: The examples of reflexive relations include:
Is equal to
Is a subset of
Divides
Is greater than or equal to
Is less than or equal to
The examples of symmetric relations include:
Is equal to
Is comparable to
... and ... are odd
The examples of symmetric relations include:
Is a subset of
Divides
implies