it is said that (a,b) is ordered pairs ,a is called first coordinate and b is called second coordinate.
let (a,b) and (c,d) are ordered pairs then
(a,b) = (c,d) if and only if a = c and b = d
(a,b) ≠ (c,d) if and only if a ≠ c and b ≠ d
cartesian product :
let A and B are two sets then cartesian product for A and B is the set of all ordered pairs (a,b) so that a ∈ A and b ∈ B ie A x B = {(a,b) :a ∈ A and b ∈ B }. If number of elements of A = n and number of elements of B = m then number of elements of A x B = n x m
let A and B are two sets then cartesian product for A and B is the set of all ordered pairs (a,b) so that a ∈ A and b ∈ B ie A x B = {(a,b) :a ∈ A and b ∈ B }. If number of elements of A = n and number of elements of B = m then number of elements of A x B = n x m
example : A ={ 1,2,3 } , B = {x , y } find A x B and B x A
numbers of elements of A = 3 and numbers of elements of B = 2 then numbers of elements of A x B and
B x A = 3 x 2 = 6 elements
A x B = { (1,x),(1,y),(2,x),(2,y),(3,x),(3,y) }
B x A = { (x,1),(x,2),(x,3),(y,1),(y,2),(y,3)}
basic property :
1 - A x ∅ = ∅
to prove this :
let ( x ,y) ∈ A x ∅ so x ∈ A and y ∈ ∅ this is contradiction so A x ∅ = ∅
2- A x B ≠ B x A
to prove this let x ∈ A x B then y ∈ B because B ≠ ∅ then (y ,x) ∈ B x A then A x B ≠ B x A
3- A x B = ∅ then A = ∅ or B = ∅
4- A x (B ∪ C) = (AxB) ∪ (AxC)
basic property :
1 - A x ∅ = ∅
to prove this :
let ( x ,y) ∈ A x ∅ so x ∈ A and y ∈ ∅ this is contradiction so A x ∅ = ∅
2- A x B ≠ B x A
to prove this let x ∈ A x B then y ∈ B because B ≠ ∅ then (y ,x) ∈ B x A then A x B ≠ B x A
3- A x B = ∅ then A = ∅ or B = ∅
4- A x (B ∪ C) = (AxB) ∪ (AxC)
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