How to prove statement: If A, then if B, then C.

Question

I was wondering. In general, what are the approaches to proving nested if statements when doing a direct proof? Thanks.

Answer 1

For "nested if statements", In general:

\begin{align} P_1\to(P_2\to\dots (P_{n-1}\to(P_n\to Q)))) &\Leftrightarrow (P_1\land P_2\land \dots \land P_{n-1}\land P_n)\to Q\tag*{$(1)$}\\ &\Leftrightarrow \neg P_1\lor\neg P_2\lor \dots \lor\neg P_{n-1}\lor\neg P_n\lor Q\tag*{$(2)$}\\ &\not\Leftrightarrow P_1\land P_2\land \dots \land P_{n-1}\land P_n\land\neg Q\tag*{$(3)$}\\ \end{align}

For $(1)$ it's a direct proof, basicly assume $P_1\dots P_n$, then prove $Q$ is a logical consequence of them. $(2)$ can also be written as: $$\neg Q\to(\neg P_1\lor\neg P_2\lor \dots \lor\neg P_{n-1}\lor\neg P_n)$$

In this case, assume $\neg Q$, if this implies any one in $\neg P_1\dots\neg P_n$ hold, it proves the statements.

Sometimes if it's hard to construct a direct proof, $(3)$ would also be a good choice, which is the negation of $(1)$, first assume that $P_1\dots P_n$, and $\neg Q$. If you can prove it's false, then that implies $(1)$ is true.

Answer 2

Note that $a \rightarrow (b \rightarrow c) \Leftrightarrow (a \wedge b) \rightarrow c$ by the exportation law. The following are common ways to prove such a statement:

Direct Proof

Assume $a$ and $b$, and then derive $c$.

Proof by Contrapositive

Prove the contrapositive of $(a \wedge b) \rightarrow c$, which is $\neg c \rightarrow \neg(a \wedge b) \Leftrightarrow \neg c \rightarrow (\neg a \vee \neg b)$. You can do this by assuming $\neg c$ and then deriving $\neg a \vee \neg b$.

Proof by Contradiction

Assume the negation of $(a \wedge b) \rightarrow c$, which is $a \wedge b \wedge \neg c$, and then derive a contradiction.

Answer 3

Assuming the 'inside' conditional is the consequent of the 'outside' conditional, you can do a conditional proof inside a conditional proof. Formally, it would look like: