Question

Let A and B be two sets such that $n\left( A \right)=3$ and $n\left( B \right)=2$. If $\left( x,1 \right)$ , $\left( y,2 \right)$ , $\left( z,1 \right)$ are in $A\times B$ , write A and B?

Answer 1

These types of problems are pretty straight forward and are very simple to solve. To handle these problems effectively, we need to have a fair amount of idea regarding set theory and all its various theorems, properties and applications. We must have in mind the terms like cardinality, cross product, union, intersections and many more. In this problem we first need to determine what will be the total size of $A\times B$ . After this, we need to write all the given terms in this set and then taking this into consideration we can easily find out the individual sets A and B.

Complete step by step solution:
Now we start off with the solution to the given problem by writing that,
The total number of elements in the set $A\times B$ will be equal to the product of the number of elements individually in set A and in set B. Mathematically we can write that the number of elements will be equal to, $n\left( A \right)\times n\left( B \right)=3\times 2=6$ terms. Now we consider the elements in $A\times B$ , they are $\left( x,1 \right)$ , $\left( y,2 \right)$ and $\left( z,1 \right)$ . We can clearly say from this that the first term of the pairs comes from set A and the second term comes from set B. So we can easily write that the set A will be equal to A=\left\\{ x,y,z \right\\} and set B will be equal to B=\left\\{ 1,2 \right\\} .This satisfies our problem.

Note: For problems like these we need to have an in depth knowledge and concept of set theory and we must be familiar with the terms of sets. We must be able to figure out all the terms that shall be present in the set $A\times B$ . From this we can easily predict the elements present individually in set A and set B. We must be careful in determining this, because we need to observe how the pairs are arranged in the set $A\times B$ .