Question

If $A = \{ - 1,1\} $ , find $A \times A \times A$

Answer 1

Hint: In order to answer this question, to multiply the given set $A$ 3 times by itself, we will go through step by step to solve it, by multiplying the given set by itself i.e.. $A \times A$ and then again we will multiply the new set $A \times A$ by the set $A$ to find the final required set.

Complete step by step solution:
As we can see that the given question is related to the topic: Relation and Function.
So,
The given set is $A = \{ - 1,1\} $ .
Now, we will multiply the given set by once itself:
$
  \therefore A \times A = \{ - 1,1\} \times \{ - 1,1\} \\
   \Rightarrow A \times A = \{ ( - 1, - 1),( - 1,1),(1, - 1),(1,1)\} \;
 $
Again, we will multiply the set $A \times A$ by the set $A$ i.e.. $A \times A \times A$ :
$
  \therefore A \times A \times A = \{ ( - 1, - 1),( - 1,1),(1, - 1),(1,1)\} \times ( - 1,1) \\
   = \{ ( - 1, - 1, - 1),( - 1, - 1,1),( - 1,1, - 1),( - 1,1,1),(1, - 1, - 1),(1, - 1,1),(1,1, - 1)(1,1,1)\} \;
 $
Hencec, the the required set $A \times A \times A$ is $\{ ( - 1, - 1, - 1),( - 1, - 1,1),( - 1,1, - 1),( - 1,1,1),(1, - 1, - 1),(1, - 1,1),(1,1, - 1)(1,1,1)\} $ .
So, the correct answer is “$\{ ( - 1, - 1, - 1),( - 1, - 1,1),( - 1,1, - 1),( - 1,1,1),(1, - 1, - 1),(1, - 1,1),(1,1, - 1)(1,1,1)\} $ ”.

Note: As we know that the given question belongs to the topic Relation and Function, so a question arises is that how a relation is related to a function? Because, all relations are functions, but not all functions are relations. Consider the case when \[SetX{\text{ }}and{\text{ }}SetY\] are coupled in such a way that all of \[SetX's\] elements are connected to exactly one element of Set Y or several of \[SetX's\] elements are related to one element of \[SetX's\] . As a result, this form of relationship is referred to as a function.