Question: A relation on the set \[A = [x:\left| x \right| < 3,x \in Z]\] where \[Z\], the set of integers, is ...

Question

A relation on the set $A = [x:\left| x \right| < 3,x \in Z]$ where $Z$ , the set of integers, is defined by $R = [(x,y):y = \left| x \right|,x \ne - 1]$ . Then the number of elements in the power set of R is:
A. $32$
B. $16$
C. $8$
D. $64$

Answer 1

Solution

We find the elements of set A using the general form of elements given in the set and then define the relation R. We can write the elements of set R as the elements from A X A and exclude the value -1 from A while writing the relation R. In the end we calculate the number of elements of R which we substitute in the formula of number of elements in the power set to obtain our answer.

Complete step-by-step answer:
We have the set $A = [x:\left| x \right| < 3,x \in Z]$
We see all elements of the set A are such that the modulus of the element is less than 3 and the elements are from the set of integers. We can write set A in form of its elements as
$A = [ - 2, - 1,0,1,2]$
Now we have a relation R defined on the set A is given by $R = [(x,y):y = \left| x \right|,x \ne - 1]$
Where the elements of R are an ordered pair $(x,y)$ where x belongs to set A and y is a function of x.
So, we can write $R \subseteq A \times A$
Now, from the condition in the relation R, $x \ne - 1$ , so we remove the element $x = - 1$ from the set A and then write $A \times A$
Now the set becomes $A = [ - 2,0,1,2]$
So, x belongs to the set $[ - 2,0,1,2]$
The elements of R are given by $R = [(x,y):y = \left| x \right|,x \ne - 1]$
So, $R = [( - 2,2),(0,0),(1,1),(2,2)]$
Number of elements of R is 4,
$\left| R \right| = 4$
We know that number of elements of power set is given by $\left| {P(R)} \right| = {2^{\left| R \right|}}$ , where P(R) is the power set of R.
Substitute the value of $\left| R \right| = 4$ in the formula.

\left| {P(R)} \right| = {2^4} \\\ \left| {P(R)} \right| = 16 \\\

So, the correct answer is “Option B”.

Note: Students many times get confused while writing the relation R because they think the values inside the modulus are positive so we will take pairs of only positive values but that is wrong because for ordered pair $(x,y)$ we are taking the value x from integers in the set A.