Let R be the set of real numbers
A = {(x, y) $\in$ R $\times$ R : y - x is an integer} is an equivalence relation on R.
B = {(x, y) $\in$ R $\times$ R : x = $\alpha$y for some rational number $\alpha$} is an equivalence relation on R.
- Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1
- Statement-1 is true, Statement-2 is false
- Statement-1 is false, Statement-2 is true
- Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement -1
The Correct Option is B
Approach Solution - 1
$\because$ x - x = 0 is an integer $\Rightarrow$ A is reflexive.
Let x - y is an integer
$\Rightarrow$ y - x is an integer
$\Rightarrow$ A is symmetric
Let x - y, y - z are integers
$\Rightarrow$ x - y + y - z is also an integer
$\Rightarrow$ x - z is an integer
$\Rightarrow$ A is transitive
$\therefore$ A is an equivalence relation.
Hence statement 1 is true.
Also B can be considered as
xBy if $\frac{x}{y} = \alpha$, a rational number
$\because \, \frac{x}{x} = 1$ is a rational number
$\Rightarrow$ B is reflexive
But $\frac{x}{y} = \alpha$ , a rational number need not imply $\frac{y}{x} = \frac{1}{\alpha}$, a rational number because
$\frac{0}{1}$ is rational $\Rightarrow \, \frac{1}{0}$ is not rational
$\therefore$ B is not an equivalence relation.
Approach Solution -2
y-x is an integer
x-x=0 is also an integer
\(\therefore\) A is reflexive
x-y=integer \(\therefore\)x-x=0
\(\therefore\) it is symmetric
x-y and y-x both are integer and sum of them is also integer
Hence, transitive.
So, it has an equivalence relation
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".