# simplify using laws and axiom of logic [closed]

$$(¬a∨b)∧(a∨b)∧¬a$$

So I have been looking at this question all day and and i have no idea how to start.

Can someone please help with me proving this algebra logic and what laws I would need to use?

I have also looked through already answered questions, and could not find anything similar.

Simplify the following statement using the laws and axioms of logic

## closed as off-topic by Graham Kemp, DMcMor, Yanior Weg, Leucippus, GNUSupporter 8964民主女神 地下教會Apr 22 at 23:06

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Graham Kemp, DMcMor, Yanior Weg, Leucippus, GNUSupporter 8964民主女神 地下教會
If this question can be reworded to fit the rules in the help center, please edit the question.

• Hello. That's not an equation, it's a formula of propositional logic, and it's not tautological, therefore it can't be proven. Also: I'm not sure what you mean by "simplify"; do you mean having a logically equivalent formula with only one logical operator? – Simone Apr 21 at 11:53
• Use the rules of commutation, absorption, and redundancy. – Graham Kemp Apr 21 at 12:08
• Thank you for helping me. but for me to understand this can you please show me the step thanks – Daisoh Apr 22 at 5:52

This expression is a "formula" because it has a well-defined mathematical value (for each allowable set of values for the variables in it). It is a "statement" because that mathematical value is either "true" or "false". But an "equation" is a statement claiming that two expressions are equal to each other (equal $$\equiv$$ equation), and there is no "=" sign in this statement. Therefore it is not an "equation".
Examine when your statement will be true. It has three parts joined together by ands: $$(\lnot a\lor b)\\(a\lor b)\\\lnot a$$
For the whole expression to be true, all three must be true. But the last one requires $$a$$ to be false. And if $$a$$ is false, the first will be true, regardless of $$b$$. Finally $$a\lor b$$ requires either $$a$$ or $$b$$ to be true. But we already know $$a$$ has to be false, so $$b$$ has to be true.
Therefore, this will be true only if $$a$$ is false and $$b$$ is true.