If A subset= C, B subset= D, show A X B subset= C X D

[osamamansoor2016]

Can some one please help me to solve these question using SETS (de Morgan's law)

\(\displaystyle \mbox{Show that, if }\, A,\, B,\, C,\, \mbox{ and }\, D\, \mbox{ are sets such that}\)

. . .\(\displaystyle A\, \subseteq\, C\, \mbox{ and }\, B\, \subseteq\, D\)

\(\displaystyle \mbox{then}\)

. . .\(\displaystyle A\, \times\, B\, \subseteq\, C\, \times\, D\)

 

[stapel]

Can some one please help me to solve these question using SETS (de Morgan's law)

\(\displaystyle \mbox{Show that, if }\, A,\, B,\, C,\, \mbox{ and }\, D\, \mbox{ are sets such that}\)

. . .\(\displaystyle A\, \subseteq\, C\, \mbox{ and }\, B\, \subseteq\, D\)

\(\displaystyle \mbox{then}\)

. . .\(\displaystyle A\, \times\, B\, \subseteq\, C\, \times\, D\)

Do the instructions specify the use of de Morgan's laws? Which laws have you been given? How far have you gotten in applying them? Where are you stuck?

Please be complete. Thank you!

 

[pythagorean314159]

de morgans laws

Can some one please help me to solve these question using SETS (de Morgan's law)

\(\displaystyle \mbox{Show that, if }\, A,\, B,\, C,\, \mbox{ and }\, D\, \mbox{ are sets such that}\)

. . .\(\displaystyle A\, \subseteq\, C\, \mbox{ and }\, B\, \subseteq\, D\)

\(\displaystyle \mbox{then}\)

. . .\(\displaystyle A\, \times\, B\, \subseteq\, C\, \times\, D\)

Am I safe to assume that A x B and C x D are Cartesian products?

 

[pka]

Can some one please help me to solve these question using SETS (de Morgan's law)
\(\displaystyle \mbox{Show that, if }\, A,\, B,\, C,\, \mbox{ and }\, D\, \mbox{ are sets such that}\)
. . .\(\displaystyle A\, \subseteq\, C\, \mbox{ and }\, B\, \subseteq\, D\)

\(\displaystyle \mbox{then}\)
. . .\(\displaystyle A\, \times\, B\, \subseteq\, C\, \times\, D\)

\(\displaystyle \begin{align*}\text{If }(s,t)\in A\times B \text{ then }&s\in A~\wedge~t\in B\\\text{by the subset definition }&s\in C~\wedge~t\in D\\&\therefore~(s,t)\in C\times D.\end{align*}\)