let-A-and-C-be-two-none-empety-set-prove-that-A-B-C-D-iff-A-C-B-D-help-me-sir-

Question Number 110219 by mohammad17 last updated on 27/Aug/20

l e t A a n d C b e t w o n o n e e m p e t y s e t p r o v e t h a t

A \subseteq B \land C \subseteq D i f f A \times C \subseteq B \times D ?

h e l p m e s i r

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

i f A \subseteq B \land C \subseteq D \Rightarrow A \times C \subseteq B \times D

o t h e r w i s e t h e r e e x i s t (a, c) \in A \times C (a \in A, c \in C) s u c h t h a t

(a, c) \in B \times D s o a \notin B o r c \notin D

i f a \notin B t h e n A ⊈ B a n d i f c \notin D t h e n C ⊈ D

t h a t i s a c o n t r u d i c t i o n .

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

i f A \times C \subseteq B \times D t h e n A \subseteq B \land C \subseteq D

o t h e r w i s e A ⊈ B o r C ⊈ D

i f A ⊈ B t h e n t h e r e e x i s t a \in A s . t a \notin B s o

f o r e a c h c \in C w e h a v e (a, c) \in A \times C a n d

(a, c) \notin B \times D t h a t i s c o n t r u d i c t i o n w i t h h y p o t h e s i s .

s i m i l a r l y w e c a n p r o v e f o r C ⊈ D .

Commented by mohammad17 last updated on 28/Aug/20

t h a n k y o u s i r