# Compound Proposition Simplification

#### User40405

##### New member
Hi all

I need to complete this question for an assignment, but I cannot seem to understand how to simplify the compound proposition with logical equivalences. If anyone here understands how to complete this question, please could you show me how, as it would be greatly appreciated. Thank you.

Here is the question:

#### steenis

##### Well-known member
MHB Math Helper
You know that $a \to b$ is equivalent to $\neg a \vee b$

Therefore $[(p \vee q) \wedge \neg p] \to q$ is equivalent to $\neg [(p \vee q) \wedge \neg p] \vee q$

Now you can simplify the last expression.

#### User40405

##### New member
You know that $a \to b$ is equivalent to $\neg a \vee b$

Therefore $[(p \vee q) \wedge \neg p] \to q$ is equivalent to $\neg [(p \vee q) \wedge \neg p] \vee q$

Now you can simplify the last expression.
Thank you so so much!

I have been checking guides the entire day and yesterday. I can now get to where you got with it, but I cannot simplify the last expression (pvq). I do not know how to change this and get the simplified form.

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Because (pvq) is equivalent to (qvp). But how does that help me?

#### Olinguito

##### Well-known member
Or use the distributive law in the first block:
$$(p\vee q)\wedge\neg p\ \equiv\ (p\wedge\neg p)\vee(q\wedge\neg p)\ \equiv\ q\wedge\neg p.$$

#### steenis

##### Well-known member
MHB Math Helper
I. study the theory

II. use $\neg (a \wedge b)$ is equivalent with $\neg a \vee \neg b$