(¬A ∨ B) is equivalent to B?

Question

Is there any proof that states (¬A ∨ B) is equivalent to B? There was an example in my text book that had a step I didn't understand. It stated (¬A ∨ B) ∧ A → B is equivalent to B ∧ A → B. I don't see how they made that step. Thank you.

Answer 1

They did not make that step. To show that $(\neg A\lor B)\land A\to B$ is equivalent to $(B\land A)\to B$, one must simply show that $(\neg A\lor B)\land A$ is equivalent to $B\land A$. Note that $$ (\neg A\lor B)\land A\equiv(\neg A\land A)\lor(B\land A)\equiv\mathbf{F}\lor(B\land A)\equiv B\land A, $$ thereby establishing your result and hence your proof of equivalence.

Concerning the relationship between $\neg A\lor B$ and simply $B$, note that the truth of $B$ guarantees the truth of $\neg A\lor B$. In this sense, we may say that $B$ is stronger than $\neg A\lor B$, but that is about it. They are certainly not equal (for instance, let $A$ and $B$ both be false, and you will obtain a counterexample).

Answer 2

It is not true that $\neg A \vee B\equiv B$, without added assumptions. However, if we assume that $A$ is true (such as in the full expression), then the $\neg A$ cannot possibly hold.

Answer 3

If (either I'm not wearing a hat, OR my hair is brown), AND I'm wearing a hat, then my hair is brown.

If my hair is brown AND I'm wearing a hat, then my hair is brown.

Examples often help to clarify.