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 $B$,

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

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 $A\in C$.

Assume that $P\left(A\right)=P\left(B\right).$ Show that $A=B.$

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

Let $A, B$ and $C$ be the sets such that $A\cup B=A\cup C$ and $A\cap B=A\cap C.$ Show that $B=C.$

Decide, among the following sets, which sets are subsets of one and another:
$A=\left\{x:x\in R \mathrm{a}\mathrm{n}\mathrm{d} x \mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{y} {\mathrm{x}}^2-8x+12=0\right\}, B=\left\{2, 4, 6\right\}, C=\left\{2, 4, 6, 8\dots \right\}, D=\left\{6\right\}.$