# 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

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• 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

## 1 Answer

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.

• hi can you please help me this question clarify and simplify. what law i need to use to simplify. – Daisoh Apr 22 at 5:51
• No. I abhor it when people would prefer to mindlessly apply mechanical methods rather than think about what the mathematics actually means. I walked you the simplification and explained exactly why it is the case. That is all I intend to do. – Paul Sinclair Apr 23 at 0:00