JEE Main & Advanced Mathematics Sets Cartesian Product of Sets

Cartesian product of sets : Let A and B be any two non-empty sets. The set of all ordered pairs \[(a,b)\] such that \[a\in A\] and \[b\in B\] is called the cartesian product of the sets A and B and is denoted by \[A\times B\].

Thus, \[A\times B=[(a,\,\,b)\,:\,\,a\in A\] and \[b\in B]\]

If \[A=\phi \] or \[B=\phi ,\] then we define \[A\times B=\phi \].

Example : Let \[A=\{a,\,\,b,\,\,c\}\] and \[B=\{p,\,q\}\].

Then \[A\times B=\{(a,\,p),\,(a,\,q),\,(b,\,p),\,(b,\,q),\,(c,\,p),\,(c,\,q)\}\]

Also \[B\times A=\{(p,\,a),\,(p,\,b),\,(p,\,c),\,\,(q,\,\,a),\,\,(q,\,\,b),\,\,(q,\,\,c)\}\]

Important theorems on cartesian product of sets :

Theorem 1 : For any three sets A, B, C

(i)  \[A\times (B\cup C)=(A\times B)\cup (A\times C)\]

(ii) \[A\times (B\cap C)=(A\times B)\cap (A\times C)\]

Theorem 2 :  For any three sets A, B, C 

\[A\times (B-C)=(A\times B)-(A\times C)\]

Theorem 3 :  If A and B are any two non-empty sets, then

\[A\times B=B\times A\Leftrightarrow A=B\]

Theorem 4 :  If \[A\subseteq B,\] then \[A\times A\subseteq (A\times B)\cap (B\times A)\]

Theorem 5 :  If \[A\subseteq B,\] then \[A\times C\subseteq B\times C\] for any set C.

Theorem 6 :  If \[A\subseteq B\] and \[C\subseteq D,\] then \[A\times C\subseteq B\times D\]

Theorem 7 :  For any sets A, B, C, D

\[(A\times B)\cap (C\cup D)=(A\cap C)\times (B\cap D)\]

Theorem 8 :  For any three sets A, B, C

(i)  \[A\times (B'\times C')'=(A\times B)\cap (A\times C)\]

(ii) \[A\times (B'\cap C')'=(A\times B)\cup (A\times C)\]