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

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