Question

If A = {1,3,5}and B = {2,3} find AXB and BXA, show that AXB = BXA
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Answer 1

SOLUTION

GIVEN

A = {1,3,5} and B = {2,3}

TO DETERMINE

• A × B , B × A

• To show : A × B ≠ B × A

CONCEPT TO BE IMPLEMENTED

Set :

A set is a well defined collection of distinct objects of our perception or of our thought to be conceived as a whole

Cartesian Product :

Let A and B are two sets. Then the Cartesian product of A and B is denoted by A × B and defined as

$\sf{A \times B = \{(x, y) : x \in A \: \: and \: \: y \in B \}}$

Equal Set :

Two sets A & B is said to be equal if every element of A is an element of B and every element of B is also an element of A

EVALUATION

Here the given sets are

A = {1,3,5} and B = {2,3}

Now

A × B = {(1,2),(3,2),(5,2),(1,3),(3,3),(5,3)}

B × A = {(2,1),(2,3),(2,5),(3,1),(3,3),(3,5)}

Thus we see that not every element of A × B is an element of B × A and not every element of B × A is an element of A × B

So A × B and B × A are not equal sets

A × B ≠ B × A

Hence proved

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Answer 2

A × B =  { (1 , 2)( 1, 3) (3, 2) (3 , 3) (5 , 2) (5,3) }

B × A  = {(2, 1) (2, 3) (2, 5) (3 ,1) (3 , 3) (3, 5) }

Step-by-step explanation:

Definition:

• If A and B. are two non empty sets, then the set of all ordered pairs

        (a, b) such that a ∈ A, b∈ B is called the Cartesian product of A and B      

         and is denoted by A × B.

• Thus A × B ={ (a, b) a ∈A , b ∈ B

read as A cross B.

• A × B is the set of all possible ordered Pairs between the elements of A and B such that the first coordinate is an elements of A and the second coordinate is an elements of B.

• B × A is the set of all possible ordered pairs between the elements of A and B such that the first coordinate is an element of B and the second coordinate is an element of A.

Given A = (1, 3, 5)    B = (2, 3)

By this

A × B =  { (1 , 2)( 1, 3) (3, 2) (3 , 3) (5 , 2) (5,3) }

B × A  = {(2, 1) (2, 3) (2, 5) (3 ,1) (3 , 3) (3, 5) }