Question Number 110219 by mohammad17 last updated on 27/Aug/20
$${let}\:{A}\:{and}\:{C}\:{be}\:{two}\:{none}\:{empety}\:{set}\:{prove}\:{that} \\ $$$${A}\subseteq{B}\:\wedge\:{C}\:\subseteq{D}\:{iff}\:{A}×{C}\:\subseteq{B}×{D}? \\ $$$${help}\:{me}\:{sir} \\ $$
Commented by kaivan.ahmadi last updated on 27/Aug/20
$${if}\:{A}\subseteq{B}\wedge{C}\subseteq{D}\Rightarrow{A}×{C}\subseteq{B}×{D} \\ $$$${otherwise}\:{there}\:{exist}\:\left({a},{c}\right)\in{A}×{C}\left({a}\in{A},{c}\in{C}\right){such}\:{that} \\ $$$$\left({a},{c}\right)\in{B}×{D}\:{so}\:{a}\notin{B}\:{or}\:{c}\notin{D} \\ $$$${if}\:{a}\notin{B}\:{then}\:{A}\nsubseteq{B}\:{and}\:{if}\:{c}\notin{D}\:{then}\:{C}\nsubseteq{D} \\ $$$${that}\:{is}\:{a}\:{contrudiction}. \\ $$
Commented by kaivan.ahmadi last updated on 27/Aug/20
$${if}\:{A}×{C}\subseteq{B}×{D}\:{then}\:{A}\subseteq{B}\wedge{C}\subseteq{D} \\ $$$${otherwise}\:{A}\nsubseteq{B}\:{or}\:{C}\nsubseteq{D} \\ $$$${if}\:{A}\nsubseteq{B}\:{then}\:{there}\:{exist}\:{a}\in{A}\:{s}.{t}\:{a}\notin{B}\:{so} \\ $$$${for}\:{each}\:{c}\in{C}\:{we}\:{have}\:\left({a},{c}\right)\in{A}×{C}\:{and} \\ $$$$\left({a},{c}\right)\notin{B}×{D}\:{that}\:{is}\:{contrudiction}\:{with}\:{hypothesis}. \\ $$$${similarly}\:{we}\:{can}\:{prove}\:{for}\:{C}\nsubseteq{D}. \\ $$$$ \\ $$
Commented by mohammad17 last updated on 28/Aug/20
$${thank}\:{you}\:{sir} \\ $$