Propositional Formulas. Is my intuition regarding this problem correct? I've been having trouble with proving whether this propositional formula is true or false.
Let's say we the following propositional formulas: $(A \wedge B), (A \wedge C), (C \wedge B)$ where each of is satisfiable and each of A, B, C is a propositional formula.Does this mean that $((A \wedge B)\wedge C)$ is also satisfiable? Now just based on my intuition I would think that this should be true. I've tried to come up with any counter arguments but no avail. Would my intuition be right here?
 A: 
Let's say we the following propositional formulas: (A∧B),(A∧C),(C∧B)
where each of is satisfiable and each of A, B, C is a propositional
formula.

You don't mention whether they are simultaneously satisfiable, so I assume not. In that case, one can have $A\wedge B\wedge C$ not satisfiable: take propositional variables $p$ and $q$ and define
$$A=p, \quad B=q \quad\text{and}\quad C=\neg p\vee\neg q.$$
If $p$ is true, but $q$ is false, $A$ and $C$ are true and then $A\wedge C$ is satisfied. If $p$ and $q$ are true, $A$ and $B$ are true and $A\wedge B$ is satisfied. And if $q$ is true but $p$ is false, $B$ and $C$ are true, and $B\wedge C$ is satisfied.
But $A\wedge B\wedge C$ can not be satisfied: if $A$ and $B$ are true, $p$ and $q$ are true and therefore $C$ is false (obs.:the first example I cooked up had three variables and dozens of formulas, this one is much better!).
If, however, $A\wedge B$, $B\wedge C$ and $A\wedge C$ are simultaneously satisfiable, as Mauro ALLEGRANZA mentions in a comment, $A\wedge B\wedge C$ is satisfiable; actually, it is enough to assume any two among $A\wedge B$, $B\wedge C$ and $A\wedge C$ are simultaneously satisfiable.
