# Ambiguous logic in Theorem statements

Whenever I have a proposition to prove such as this:

$$f:X\rightarrow Y \text{ continuous, X connected} \implies Y \text{ connected}$$

I get confused whether the following two are equivalent to the above or not (noting the brackets):

$$[f:X\rightarrow Y \text{ continuous, X connected} \implies Y \text{ connected}]$$

$$f:X\rightarrow Y \text{ continuous, then [X connected} \implies Y \text{ connected}]$$

The problem particularly arises when I take the contrapositive of such a statement. So which one is correct?

• All three are equivalent. – Kavi Rama Murthy Apr 18 at 7:17

Both are correct. In terms of Logic, the first formulation is$$(a\wedge b)\implies c,\tag1$$whereas the second one is$$a\implies(b\implies c).\tag2$$But $$(1)$$ and $$(2)$$ are equivalent.