Mathematical Logic

Question

prove $(¬A→¬B)∧(A→C)→(B→C)$

Answer 1

By using the contrapositive, we can write the LHS as:

$$(B \rightarrow A) \land (A \rightarrow C)$$

This says that if $x \in B$, then $x \in A$ and if $x \in A$, then $x \in C$. Both of them hold true together.

We want to prove that if $x \in B$, then $x \in C$.

Let us assume the LHS is true.

Now, if $x \in B$, then by LHS we know $x \in A$. Now since $x \in A$, $x \,\text{also} \in C$, as that's what we have assumed to be true.

This shows that if $x \in B$, then $x \in C$.

You can look at it as a sort of transitive relation.