Mathematical Reasoning and Writing:

[mliuzzolino]

Homework Statement

Let f : A → B be a function and let S, T $\subseteq$ A and U, V $\subseteq$ B:

Prove that if U $\subseteq$ V; then f^-1(U) $\subseteq$ f^-1(V):

Homework Equations

Preimage: f^-1(U) = {x $\in$ f^-1(U) $\ni$ f(x) $\subseteq$ U}

The Attempt at a Solution

Assume U $\subseteq$ V.

Let x $\in$ f^-1(U), then f(x) = y $\subseteq$ U.

Since U $\subseteq$ V and y $\in$ U, then y $\in$ V.

Then by f^-1(V), for y $\in$ V, x $\subseteq$ f^-1(V).

Therefore, f^-1(U) $\subseteq$ f^-1(V).

Q.E.D.

 

[jbunniii]

The Attempt at a Solution

Assume U $\subseteq$ V.

Let x $\in$ f^-1(U), then f(x) = y $\subseteq$ U.

Should be $f(x) = y \in U$.

Since U $\subseteq$ V and y $\in$ U, then y $\in$ V.

Yes.

Then by f^-1(V), for y $\in$ V, x $\subseteq$ f^-1(V).

Right idea, but stated poorly. It is not clear what the words "by" and "for" are attempting to convey here. And again you used $\subseteq$ when you should have used $\in$.

Here is a clearer and simpler statement: $y = f(x) \in V$, so $x \in f^{-1}(V)$.

Therefore, f^-1(U) $\subseteq$ f^-1(V).

Correct.

 

[mliuzzolino]

Should be $f(x) = y \in U$.


Yes.


Right idea, but stated poorly. It is not clear what the words "by" and "for" are attempting to convey here. And again you used $\subseteq$ when you should have used $\in$.

Here is a clearer and simpler statement: $y = f(x) \in V$, so $x \in f^{-1}(V)$.


Correct.

The part you clarified was the part I was really struggling with. Thanks a lot!