Properties of Cartesian product of Sets

For the sets A, B, C and D, we have-
Property 1: \(\ A\times(B\cup C)=(A\times B)\cup (A\times C) \)
Proof:
Let \(\ (x,y)\in A\times (B\cup C)\)
\(\ \Leftrightarrow x\in A, y\in (B\cup C) \)
\(\ \Leftrightarrow x\in A, (y\in B \) or \(\ y\in C ) \)
\(\ \Leftrightarrow (x\in A, y\in B) \) or \(\ (x\in A, y\in C) \)
\(\ \Leftrightarrow (x,y)\in A\times B \) or \(\ (x,y)\in A\times C \)
\(\ \Leftrightarrow (x,y)\in (A\times B) \cup (A\times C) \)
⸫ \(\ A\times(B\cup C)=(A\times B)\cup (A\times C) \)


Property 2: \(\ A\times(B\cap C)=(A\times B)\cap (A\times C) \)
Proof:
Let \(\ (x,y)\in A\times (B\cap C)\)
\(\ \Leftrightarrow x\in A, y\in (B\cap C) \)
\(\ \Leftrightarrow x\in A, (y\in B \) and \(\ y\in C ) \)
\(\ \Leftrightarrow (x\in A, y\in B) \) and \(\ (x\in A, y\in C) \)
\(\ \Leftrightarrow (x,y)\in A\times B \) and \(\ (x,y)\in A\times C \)
\(\ \Leftrightarrow (x,y)\in (A\times B) \cap (A\times C) \)
⸫ \(\ A\times(B\cap C)=(A\times B)\cap (A\times C) \)


Property 3: \(\ (A\times B)\cup (C\times D)=(A\cup C)\times (B\cup D) \)
Proof:
Let \(\ (x,y)\in (A\times B)\cup (C\times D) \)
\(\ \Leftrightarrow (x,y)\in (A\times B) \) or \(\ (x,y)\in (C\times D) \)
\(\ \Leftrightarrow (x\in A, y\in B \)) or \(\ (x\in C, y\in D) \)
\(\ \Leftrightarrow (x\in A \) or \(\ x\in C \)), \(\ (y\in B \) or \(\ y\in D) \)
\(\ \Leftrightarrow (x\in A\cup C \)), \(\ (y\in B\cup D) \)
\(\ \Leftrightarrow (x,y)\in (A\cup C)\times (B\cup D) \)
⸫ \(\ (A\times B)\cup (C\times D)=(A\cup C)\times (B\cup D) \)


Property 4: \(\ (A\times B)\cap (C\times D)=(A\cap C)\times (B\cap D) \)
Proof:
Let \(\ (x,y)\in (A\times B)\cap (C\times D) \)
\(\ \Leftrightarrow (x,y)\in (A\times B) \) and \(\ (x,y)\in (C\times D) \)
\(\ \Leftrightarrow (x\in A \) and \(\ x\in C \)) and \(\ (y\in B \) and \(\ y\in D) \)
\(\ \Leftrightarrow (x\in A\cap C \)) and \(\ (y\in B\cap D) \)
\(\ \Leftrightarrow (x,y)\in (A\cap C)\times (B\cap D) \)
⸫ \(\ (A\times B)\cap (C\times D)=(A\cap C)\times (B\cap D) \)


Property 5: \(\ A\times(B-C)=(A\times B)-(A\times C) \)
Proof:
Let \(\ (x,y)\in A\times (B-C) \)
\(\ \Leftrightarrow x\in A \) and \(\ y\in (B-C) \)
\(\ \Leftrightarrow x\in A \) and \(\ (y\in B \) and \(\ y\notin C) \)
\(\ \Leftrightarrow ( x\in A \) and \(\ y\in B ) \) and \(\ (x\in A \) and \(\ y\notin C) \)
\(\ \Leftrightarrow ( x,y) \in A\times B \) and \(\ (x,y)\notin A\times C \)
\(\ \Leftrightarrow (x,y)\in (A\times B) -(A\times C) \)
⸫ \(\ A\times(B-C)=(A\times B)-(A\times C) \)


Property 6: \(\ (A-B)\times C=(A\times C)-(B\times C) \)
Proof:
Let \(\ x\in (A-B)\times C \)
\(\ \Leftrightarrow x\in (A-B) \) and \(\ y\in C \)
\(\ \Leftrightarrow (x\in A \) and \(\ x\notin B) \) and \(\ y\in C \)
\(\ \Leftrightarrow ( x\in A \) and \(\ y\in C ) \) and \(\ (x\notin B \) and \(\ y\in C) \)
\(\ \Leftrightarrow ( x,y) \in A\times C \) and \(\ (x,y)\notin B\times C \)
\(\ \Leftrightarrow (x,y)\in (A\times C) -(B\times C) \)
⸫ \(\ (A-B)\times C=(A\times C)-(B\times C) \)


Property 7: \(\ A\times B = B\times A \) iff \(\ A=\phi \) or \(\ B=\phi \) or \(\ A=B \)
Proof:
Case 1:
Let \(\ A=\phi \)
Now \(\ A\times B = \phi \times B = \phi\)   [⸪ \(\ A=\phi \) ]
also \(\ B\times A = B\times \phi =\phi \)   [⸪ \(\ A=\phi \) ]
Hence \(\ A\times B = B\times A \)
Case 2:
Let \(\ B=\phi \)
Now \(\ A\times B = A\times \phi =\phi \)   [⸪ \(\ B=\phi \)]
also \(\ B\times A = \phi \times A =\phi \)   [⸪ \(\ B=\phi \)]
Hence \(\ A\times B = B\times A \)
Case 3:
Let \(\ A=B \)
Now \(\ A\times B = A\times A \)   [⸪ \(\ A=B \) ]
also \(\ B\times A = A\times A \)   [⸪ \(\ A=B \) ]
Hence \(\ A\times B = B\times A \)


Q. If \(\ A\times B=B\times A \) then show that \(\ A=B \)
Solution:
Let \(\ a\in A \Rightarrow (a,b)\in A\times B ~\forall~ b\in B \)
    \(\ \Rightarrow (a,b)\in B\times A \)   [⸪ \(\ A\times B=B\times A \)]
    \(\ \Rightarrow a\in B \) [definition of Cartesian product of sets]
Conversely,
Let \(\ b\in A \Rightarrow (a,b)\in A\times B ~\forall~ a\in A \)
    \(\ \Rightarrow (a,b)\in B\times A \)   [⸪ \(\ A\times B=B\times A \)]
    \(\ \Rightarrow b\in A \) [definition of Cartesian product of sets]
Hence, \(\ A=B \)


Q. If \(\ A\times B=A\times C \) then show that \(\ B=C \)
Solution:
Let \(\ b\in B \Rightarrow (a,b)\in A\times B ~\forall~ a\in A \)
    \(\ \Rightarrow (a,b)\in A\times C \)   [⸪ \(\ A\times B=A\times C \)]
    \(\ \Rightarrow b\in C \) [definition of Cartesian product of sets]
Conversely,
Let \(\ c\in C \Rightarrow (a,c)\in A\times C ~\forall~ a\in A \)
    \(\ \Rightarrow (a,c)\in A\times B \)   [⸪ \(\ A\times B=A\times C \)]
    \(\ \Rightarrow c\in B \) [definition of Cartesian product of sets]
Hence, \(\ B=C \)


Q. If A, B and C be three sets where \(\ B\subseteq A \) then show that \(\ (B\times C)\subseteq (A\times C) \)
Solution:
Let \(\ (b,c)\in B\times C \)
\(\ \Rightarrow b\in B \) and \(\ c\in C \)
\(\ \Rightarrow b\in A \) and \(\ c\in C \)   [⸪ \(\ B\subseteq A \)]
\(\ \Rightarrow (b,c)\in A\times C \)
Hence \(\ (B\times C)\subseteq (A\times C) \)


Q. If A, B and C be three sets where \(\ A\subseteq B \) then show that \(\ (A\times C)\subseteq (B\times C) \)
Solution:
Let \(\ (a,c)\in A\times C \)
\(\ \Rightarrow a\in A \) and \(\ c\in C \)
\(\ \Rightarrow a\in B \) and \(\ c\in C \)   [⸪ \(\ A\subseteq B \)]
\(\ \Rightarrow (a,c)\in B\times C \)
Hence \(\ (A\times C)\subseteq (B\times C) \)


Q. If \(\ A\subseteq B \) and \(\ C\subseteq D \) then show that \(\ (A\times C)\subseteq (B\times D) \)
Solution:
Let \(\ (a,c)\in A\times C \)
\(\ \Rightarrow a\in A \) and \(\ c\in C \)
\(\ \Rightarrow a\in B \) and \(\ c\in D \)   [⸪ \(\ A\subseteq B \) and \(\ C\subseteq D \)]
\(\ \Rightarrow (a,c)\in B\times D \)
Hence \(\ (A\times C)\subseteq (B\times D) \)