Operations on Sets

In the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If $x\in A$ and $A\not\subset B$, then $x\in B$

In each the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If $A\not\subset B$ and $\mathrm{B}\not\subset \mathrm{C}$, then $A\not\subset \mathrm{C}$

In the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If $A\subset B$ and $B\subset C$, then $A\subset C$

If $x\in A$ and $A\in B$, then $x\in B$.

Show that the following four conditions are equivalent:
 $A\subset B$
 $A-B=\varphi$
 $A\cup B=B$
 $A\cap B=A$

Show that the following four conditions are equivalent:
 $A\subset B$
 $A-B=\varphi$
 $A\cup B=B$
 $A\cap B=A$

Show that the following four conditions are equivalent:
(I)   $A\subset B$
(ii)   $A-B=\varphi$
(iii)  $A\cup B=B$
(iv)   $A\cap B=A$

Show that for any sets $A$ and $B$,

$A=(A\cap B)\cup (A-B) \mathrm{and} A\cup (B-A)=(A\cup B)$

If $A=\{x:x$ is a natural number $\}$, $B=\{x:x$ is an even natural number } $C=\{x:x$ is an odd natural number $\}$ and $D=\{x:x$ is a prime number $\}$, then, $A\cap D$ is same as which set?

If $A=\{x:x$ is a natural number$\}$, $B=\{x:x$ is an even natural number } $C=\{x:x$ is an odd natural number $\}$ and $D=\{x:x$ is a prime number$\}$, then which set is equal to $A\cap C$?

Is it true that for any sets $A$ and $B, P\left(A\right)\cup P\left(B\right)=P(A\cup B)$? 

Show that if $A\subset B$, then $C\mathit{-}B\subset C\mathit{-}A$.

If $A\subset B$ and $x\notin B$, then $x\notin A$.

If $A\subset B$ and $B\in C$, then $A\in C$.

If $R$ is the set of real numbers and $Q$ is the set of rational numbers, then what is $R-Q?$

If $X=\left\{a,b,c,d\right\}$ and $Y=\left\{f,b,d,g\right\},$ find $X\cap Y$.

If $X=\left\{a,b,c,d\right\}$ and $Y=\left\{f,b,d,g\right\},$ find $Y-X$.

If $X=\left\{a,b,c,d\right\}$ and $Y=\left\{f,b,d,g\right\},$ find $X-Y$.

If $A=\left\{3,6,9,12,15,18,21\right\},B=\left\{4,8,12,16,20\right\},C=\left\{2,4,6,8,10,12,14,16\right\},$$D=\left\{5,10,15,20\right\};$ Find $D-C$.

If $A=\left\{3,6,9,12,15,18,21\right\},B=\left\{4,8,12,16,20\right\},C=\left\{2,4,6,8,10,12,14,16\right\},$$D=\left\{5,10,15,20\right\};$ Find $C-D$.