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

Hint: 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.