Question Number 110223 by mohammad17 last updated on 27/Aug/20
$${let}\::{A},{B},{C},{D}\:{be}\:{anone}\:{empety}\:{set}\:{prove}\:{that}\: \\ $$$${A}×{B}={C}×{D}\leftrightarrow{A}={C}\wedge{B}={D}\:? \\ $$$${help}\:{me}\:{sir} \\ $$
Commented by kaivan.ahmadi last updated on 27/Aug/20
$${I}\:{think}\:{it}\:{isnt}\:{true},\:{because}\:{in}\:{set}\:{theory}\:{we} \\ $$$${have}\:{C}\wedge{B}={C}\cap{B}. \\ $$$${Now}\:{let} \\ $$$${A}={D}={B}\cap{C}=\left\{\mathrm{2}\right\} \\ $$$${B}=\left\{\mathrm{1},\mathrm{2}\right\} \\ $$$${C}=\left\{\mathrm{2},\mathrm{3}\right\} \\ $$$${then} \\ $$$${A}×{B}=\left\{\left(\mathrm{2},\mathrm{1}\right),\left(\mathrm{2},\mathrm{2}\right)\right\} \\ $$$${C}×{D}=\left\{\left(\mathrm{2},\mathrm{2}\right),\left(\mathrm{3},\mathrm{2}\right)\right\} \\ $$$$\Rightarrow{A}×{B}\neq{C}×{D} \\ $$
Commented by mohammad17 last updated on 28/Aug/20
$${sir}\:{ther}\:{are}\:{four}\:{none}\:{empity}\:{sets}\:\left({A},{B},{C},{D}\right) \\ $$
Commented by kaivan.ahmadi last updated on 28/Aug/20
$${if}\:{A}={C}\:{and}\:{B}={D}\:{it}\:{is}\:{clear}\:{that} \\ $$$${A}×{B}={C}×{D}. \\ $$$${now}\:{if}\:{A}×{B}={C}×{D}\:{then}\:{we}\:{must}\:{prove} \\ $$$${that}\:{A}={C}\:{and}\:{aB}={D}. \\ $$$${let}\:{a}\in{A}\:{and}\:{b}\in{B}\:{then}\:{we}\:{have} \\ $$$$\left({a},{b}\right)\in{A}×{B}\:{so}\:\left({a},{b}\right)\in{C}×{D}\Rightarrow{a}\in{C}\:{and}\:{b}\in{D} \\ $$$${so}\:{A}\subseteq{C}\:{and}\:{B}\subseteq{D}. \\ $$$${similarly}\:{we}\:{can}\:{prove}\:{that}\:{C}\subseteq{A}\:{and}\:{D}\subseteq{B}. \\ $$$${so}\:{A}={C}\:{and}\:{B}={D}. \\ $$
Commented by kaivan.ahmadi last updated on 28/Aug/20
$${excuse}\:{me}\:{i}\:{didnt}\:{understand}\:{that}\:\wedge\:{means} \\ $$$$\&\:{in}\:{this}\:{question}.\:{because}\:\wedge\:{is}\:{a}\:{symbol} \\ $$$$\left({joint}\right){in}\:{latice}\:{theory}. \\ $$