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?

  • 1
    $\begingroup$ All three are equivalent. $\endgroup$ – 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.


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