How to simplify $\neg x\lor (x \wedge y)$

Question

I have this logical statement

$$\neg x\lor (x \wedge y)$$

However I do not know what is considered a valid transformation. Normally if there is an $\wedge$ in the middle I treat it like multiplication and pull out some "shared" piece but here I don't know how to use distributive properties.

Answer 1

I am familiar with those laws but not how they would be applied here. NOT always messes me up when it comes to distributive manipulations.

Practice it slowly, one step at a time, until perfect.   When distributing, carefully keep the "nots" stuck to their variables.

$$\raise{0.5ex}{\neg x\vee }(x\wedge y)$$

So both the $x$ and $y$ terms in the brackets need to get a $\neg x \vee$ appended.   That is Distribution of $\vee$ over $\wedge$.

$$(\raise{0.5ex}{\neg x\vee}x) \wedge (\raise{0.5ex}{\neg x\vee}y)$$

Now $\neg x\vee x$ is a tautology.   That is Complementation for $\vee$. $$\raise{0.25ex}{\top\wedge}(\neg x\vee y)$$

Finally, $\top\wedge A$ is $A$ for any $A$.   That is identity for $\wedge$.

$$\neg x\vee y$$

At last, that is all. With practice you will see it as $$\require{cancel}\neg x\vee(\cancel{\color{gray}{x\wedge}} y)$$

Answer 2

You can use the distributive law:

$$(\neg x ) \vee (x \wedge y) \equiv (\neg x \vee x) \wedge (\neg x \vee y)$$

Now, the statement $x \vee \neg x \equiv T$ (do you see why?)

$$T \wedge (\neg x \vee y)$$

The proposition $T \wedge p \equiv p$ (again, do you see why?)

So we have just

$$\neg x \vee y $$

Which happens to be equivalent to $$x \rightarrow y$$

Both answers should be fine, it's up to your teacher though.

Answer 3

If you write the truth table, you see a familiar pattern. Note that your expression is just $x\to y$ . We know that $x\to y\equiv \neg x \vee y$.

You can also use the distributive properties as given in the other answer.