# Using law of logic, simplify the statement form $A \lor [\neg(\neg A)\land B]$

Simplifying the given expression led me to $$A\lor\neg B$$. Here is what I did:

$$A \lor [\neg(\neg A)\land B]$$ -------> Given

$$A \lor [\neg(\neg A)\lor \neg B]$$ ------> De Morgan's Law

$$A \lor (A\lor \neg B)$$ -----------> Law of double negation

$$(A \lor A)\lor \neg B$$ -----------> Associative Law

$$A \lor \neg B$$-----------------> Idempotent Law

Now, I want to know if my application of De Morgan's Law (second step) is correct. Thanks!

• Looks good to me Jun 17, 2020 at 5:33
• You may also rewrite the last statement as $B\to A$ (but maybe you want to keep it with $\land,\lor,\lnot$ only) Jun 17, 2020 at 9:21
• Please use mathjax for mathematical formulas. eg the $\vee$ symbol is \vee, the $\neg$ symbol is \neg and the $\wedge$ is \wedge. All you have to do is to surround your formula, e.g A \vee \neg B, with dollar symbols and voilà. : $A \vee \neg B$ :) Jun 17, 2020 at 9:26
• @OlivierRoche I use \land (logic and) for $\land$ and \lor (logic or) for $\lor$. Easier to remember for me. Jun 17, 2020 at 9:44
• @HennoBrandsma Nice, I wasn't aware of these aliases :) Jun 17, 2020 at 10:20

Everything is fine, but you may simplify even further : by definition of $$\to$$, the last line is equivalent to $$B \to A$$