Question

If ${\text{A = \\{ 0,1\\} }}$and ${\text{B = \\{ 1,2,3\\} }}$, show that ${\text{A}} \times {\text{B}} \ne {\text{B}} \times {\text{A}}$

Answer 1

For any of the random given set, let ${\text{A = \\{ a\\} ,B = \\{ b\\} }}$than ${\text{A}} \times {\text{B}} = \\{ (a,b)\\}$ apply this concept in the above given question, and we can continue with the calculation of both the terms and we can show that ${\text{A}} \times {\text{B}} \ne {\text{B}} \times {\text{A}}$.

Complete step by step answer:

As per the given sets are ${\text{A = \\{ 0,1\\} }}$and ${\text{B = \\{ 1,2,3\\} }}$
Let us first calculate the term of ${\text{A}} \times {\text{B}}$,
As, if ${\text{A = \\{ a\\} ,B = \\{ b\\} }}$then ${\text{A}} \times {\text{B}} = \\{ (a,b)\\}$,
So we get,
${\text{A}} \times {\text{B}} = \\{ (0,1),(0,2),(0,3),(1,1),(1,2),(1,3)\\}$
And then calculating for ${\text{B}} \times {\text{A}}$,
${\text{B}} \times {\text{A}} = \\{ (1,0),(1,1),(2,0),(2,1),(3,0),(3,1)\\}$
Hence, from the above sets we can clearly interpret that ${\text{A}} \times {\text{B}} \ne {\text{B}} \times {\text{A}}$.
Hence, proved.

Note: A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair ${\text{(x,y)}}$ is in the relation.
1)Sets are collections of well-defined objects; relations indicate relationships between members of two sets A and B, and functions are a special type of relationship where there is exactly (or at most) one relationship for each element ${\text{a}} \in {\text{A}}$ with an element in B.
2)Relations, Cartesian product, Relation on a Set. A relation R from X to Y is a subset of the Cartesian product ${\text{X$\times$Y}}$. The domain of a relation R is the set of all the first components of the ordered pairs that constitute the relation. The range of R is the set of all the second components of every ordered pair in R.