let A and C be two none empety set prove that A⊆B ∧ C ⊆D iff A×C ⊆B×D? help me sir


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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

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