Write the LaTeX commands for the following information: Let and C be any sets. T

Question

Write the LaTeX commands for the following information:

Let and C be any sets. The objective is to prove that if A C and B C then A B C. To show A U C, it is enough to prove that Forall B, A C. By the definition of union x A or x elementof B From x elementof A and A C, x elementof C. From x elementof B and B C. x elementof C. In both the cases. x elementof C Thus, for any x, if x elementof A B then x elementof C. By the definition of subset. A B C. Therefore, the result A B C is true for all sets A, B, and C with A B and A C.

Explanation / Answer

Let $A$,$B$ and $C$ be any sets.\ The objective is to prove that if $A subseteq C$ and $B subseteq C$ then $A cup B subseteq C $ .\ To show that $A cup B subseteq C $,it is enough to prove that $orall x$,if $in A cup B$,then $x in C$.\ Let $x in A cup B$ and $A subseteq C$, $A subseteq C$\ By the definition of union $x in A$ and $x in B$\ From $xin A$ and $A subseteq C$,$x in C$\ From $xin B$ and $B subseteq C$,$x in C$\ In both cases,$x in C$\ Thus, for any x, if $x in A cup B$,then $xin C$.\ By the definition of subset,$A cup B subseteq C $ Therefore,the result $A cup B subseteq C $ is true for all sets A,B and C with $A subseteq C$ and $A subseteq C$

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