let-A-B-C-D-be-anone-empety-set-prove-that-A-B-C-D-A-C-B-D-help-me-sir-

Question Number 110223 by mohammad17 last updated on 27/Aug/20

l e t : A, B, C, D b e a n o n e e m p e t y s e t p r o v e t h a t

A \times B = C \times D \leftrightarrow A = C \land B = D ?

h e l p m e s i r

Commented by kaivan.ahmadi last updated on 27/Aug/20

I t h i n k i t i s n t t r u e, b e c a u s e i n s e t t h e o r y w e

h a v e C \land B = C \cap B .

N o w l e t

A = D = B \cap C = {2}

B = {1, 2}

C = {2, 3}

t h e n

A \times B = {(2, 1), (2, 2)}

C \times D = {(2, 2), (3, 2)}

\Rightarrow A \times B \neq C \times D

Commented by mohammad17 last updated on 28/Aug/20

s i r t h e r a r e f o u r n o n e e m p i t y s e t s (A, B, C, D)

Commented by kaivan.ahmadi last updated on 28/Aug/20

i f A = C a n d B = D i t i s c l e a r t h a t

A \times B = C \times D .

n o w i f A \times B = C \times D t h e n w e m u s t p r o v e

t h a t A = C a n d a B = D .

l e t a \in A a n d b \in B t h e n w e h a v e

(a, b) \in A \times B s o (a, b) \in C \times D \Rightarrow a \in C a n d b \in D

s o A \subseteq C a n d B \subseteq D .

s i m i l a r l y w e c a n p r o v e t h a t C \subseteq A a n d D \subseteq B .

s o A = C a n d B = D .

Commented by kaivan.ahmadi last updated on 28/Aug/20

e x c u s e m e i d i d n t u n d e r s t a n d t h a t \land m e a n s

& i n t h i s q u e s t i o n . b e c a u s e \land i s a s y m b o l

(j o i n t) i n l a t i c e t h e o r y .