# Simple question on SET theory

I have a bit difficulty :

If A,B and C are sets then:

• A ∈ B , B ⊆ C.Then A ⊆ C is true.
• A ⊄ B.Then B ⊂ C is true.
• C ∈ p(A) iff C ⊆ A, where p(A) denotes the power of set A.

All are correct?It's confusing me alot.

Basically i am a CS student, very poor in maths, but now have to do it, as maths is applicable everywhere.

Also please suggest me few tips and logics to study maths in a easy way.

Thank You

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Your first point would be correct if you either changed "$A\subseteq C$" to "$A\in C$", or changed "$A\in B$" to "$A\subseteq B$". The first change works because $B\subseteq C$ means every element of $B$ is an element of $C$, so if $A\in B$, then $A$ in $C$. The second change works because $A\subseteq B$ means every element of $A$ would be an element of $B$, so since every element of $B$ is an element of $C$, then every element of $A$ is an element of $C$, meaning $A\subseteq C$.
Your second point is incorrect. Knowing only that $A\not\subset B$ (that is, that some element of $A$ isn't in $B$) tells us nothing about whether all the elements of $B$ are all in $C$.
Your third point is correct. The $\mathcal{P}(A)$ is defined to be the set of all $C$ such that $C\subseteq A$. Thus, $C\subseteq A$ if and only if $C\in\mathcal{P}(A)$.
As stated, it is incorrect. Given $A\in B$ and $B\subseteq C$, we can only conclude that $A\in C$ (by definition of the statement "$B\subseteq C$"), and not necessarily that $A\subseteq C$. –  Cameron Buie Mar 25 '13 at 12:24