What is the contrapositive of the this statement " If $X$ is connected and $f:X\rightarrow Y$ is a continuous surjective map then Y is connected"

Question

What is the contrapositive of the this statement " If $X$ is connected and $f:X\rightarrow Y$ is a continuous surjective map then Y is connected"

I know this is an important result but what is the contrapositive of this sentence.

My efforts:

$A\implies B$

Contrapositive: $\text{not}\;B\implies \text{not}\;A$

If $f:X\rightarrow Y$ is a continuous function and $Y$ is disconnected then $X$ is disconnected.

May be I am correct but I don't know why I am not confident about the answer. I just solved it using my intuition. How to be sure about these kind of questions.

Answer 1

You really have a statement $(A\wedge B) \implies C$, where $A$ is "$X$ is connected", $B$ is "$f\colon X \to Y$ is a continuous surjective map", and $C$ is "Y is connected". Applying the contrapositive to this gives $$ \neg C \implies \neg(A\wedge B),$$ which is equivalent (you can check using truth tables) to $$ \neg C \implies (\neg A\vee \neg B).$$ Hence you get the statement "If $Y$ is not connected then $f\colon X \to Y$ is not a continuous surjective map or $X$ is not connected."

Note that "or" can be a little misleading. Both $\neg A$ and $\neg B$ can hold, but it is sufficient for only one to hold.

Answer 2

Correct me if wrong:

You have statement

1) $A:$ $X$ is connected and $f:X \rightarrow Y$

is a continuous (and) surjective map.

2) $B$: $Y$ is connected.

Given :$A \Rightarrow B$.

Contrapositive:

Not $B \Rightarrow$ not $A$.

Not $B$: $Y$ is not connected.

Not $A:$ $X$ is not connected

or $f:X \rightarrow Y$ is not continuos

or $f:X \rightarrow Y$ is not surjective.