<|im_start|>user
Find the logical equivalent of the statement: (($X$ AND $Y$) IMPLIES ($W$ OR $Z$)).<|im_end|>
<|im_start|>assistant
The logical equivalent of the statement: (($X$ AND $Y$) IMPLIES ($W$ OR $Z$)) can be found by using the logical equivalence known as the implication rule: $A \implies B$ is equivalent to $\neg A \lor B$.

Using this rule, we can convert the original statement as follows:

(($X$ AND $Y$) IMPLIES ($W$ OR $Z$)) $\equiv$ ($\neg (X \land Y)$ OR ($W$ OR $Z$))

So, the logical equivalent of the statement (($X$ AND $Y$) IMPLIES ($W$ OR $Z$)) is ($\neg (X \land Y)$ OR ($W$ OR $Z$)).<|im_end|>
<|im_start|>user
I'm not sure I understand, how did you come to this conclusion? Can explain step by step?  Also provide a detailed explanation to the logical equivalence or implication rule?<|im_end|>
<|im_start|>assistant
Certainly! Let me break it down for you.

1. The statement we’re working with is: `(X AND Y) IMPLIES (W OR Z)`

2. To find its logical equivalent, we’ll use the material implication rule. This rule states that `p IMPLIES q` is the same as `~p OR q`. This means we need to negate the antecedent (the part before “IMPLIES”) and join it with the consequent (the part after “IMPLIES”) using “OR”.

3. So, let’s start by negating the antecedent (`X AND Y`). We should use De Morgan’s laws for this: `~(X AND Y) = ~X OR ~Y`

4. Now we can join this with the consequent, `W OR Z`: `~(X AND Y) OR (W OR Z)` This is the logical equivalent of our original statement.

To further explain the material implication rule, it’s based on the idea that an implication `p IMPLIES q` is false only when p is true and q is false. In all other cases, it’s true.

For example, take the statement "If it’s raining, then the ground is wet." This statement is false only if it’s raining and the ground isn’t wet. If it isn’t raining, then the statement is true because the `if` part isn’t satisfied. Similarly, if it’s raining and the ground is wet, then the statement is true.

The material implication rule formalizes this idea by stating that `p IMPLIES q` is equivalent to "If `p` is true, then `q` must be true; otherwise, the implication is true regardless of `q`’s truth value." Symbolically:

`p IMPLIES q ≡ ~p OR q`

In other words, the implication is true unless `p` is true and `q` is false. This is captured by negating `p` (i.e., `~p`) and joining it with `q` using disjunction (i.e., `OR q`).

I hope this helps! Let me know if you have any more questions!<|im_end|>
