Question

if A ={1,2,3} and B={4,5} find A x B and B x A. Determine n(A x B) and n(B x A). could someone explain it in detail please!
if A ={1,2,3} and B={4,5} find A x B and B x A. Determine n(A x B) and n(B x A). could someone explain it in detail please!
@Mathematics

Answer 1

If you have two sets A and B, then the cartesian product of those sets, A x B, is the set consisting of all ordered pairs, (a,b), where a in A, b in B.

For example,

Answer 2

for example with your A and B,

A x B = { (1,4), (1,5), (2,4), (2,5), (3,4), (3,5) }

Answer 3

Hopefully it's clear that in general if A and B are finite sets, then

n(A x B) = n(A) x n(B)

In this case, n(A) = 3, n(B) = 2, n(A x B) = 6 and therefore the above relation holds.

Answer 4

One way to see the relation n(A x B) = n(A) x n(B) is to think of the members of A x B in a grid where
A has n members, A = {a1, a2, ..., an}
and B has m members, B = {b1, b2, ..., bm}


a1 a2 a3 .... an
b1 (a1,b1) (a2,b1) (a3,b1) .... (an,b1)
b2 (a1,b2) (a2,b2) (a3,b2) .... (an,b2)
...
bm (a1,bm) (a2,bm) (a3,bm) .... (an,bm)

Now A x B are all the members in the grid, and has n x m members.

Answer 5

Now, given all that, what are the members of B x A?

Answer 6

B x A= { (4,1), (4,2), (4,3), (5,1), (5,2), (5,3) } ?

Answer 7

Yes, exactly.

Answer 8

and what is n(B x A)?

Answer 9

n(B x A) would be 6, right ?

Answer 10

Yes.

Answer 11

Notice that n(B x A) = n(B) x n(A)
and
n(A x B) = n(A) x n(B)
= n(B) x n(A), because multiplication of integers is commutative
= n(B x A)

Hence in general

n(A x B) = n(B x A) = n(A) x n(B) (or n(B) x n(A))

Answer 12

Hence, for example, if A = {1,2,3,4,5,6,7,8,9,10} and B = {a,b,c,d,e,f,g,h, ..., z}

then n(AxB) = n(BxA) = 260

Answer 13

Also n(AxA) = 10^2 = 100

Answer 14

Make sense?

Answer 15

yeah it does. thanks !

Answer 16

great