Question

Please help me on this question :(

Answer 1

Answer: The filled out table is shown below

The main table is at the top. This is what it would look like if you filled out your blank boxes. The second section below shows scratch work to help form the more complicated columns.

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Explanation:

We have two atomic variables p and q. They can each take on values of True (T) or False (F).

There are 2*2 = 4 combinations of T and F as shown below

• T,T ... both true
• T,F ... p is true but q is false
• F,T ... p is false, but q is true
• F,F ... both are false

That takes care of the first two columns of the table. Note how there aren't any other ways to have two truth values together.

The third column ~q is where we flip everything in the q column. So if q is true, then ~q is false, and vice versa. A similar situation happens with the ~p column as well.

The ~p v q column is where we apply a disjunction to ~p with q. The V stands for "or". It means either ~p is true or q is true. If either are true, then (~p v q) is true. Otherwise it's false. Put another way, (~p v q) is only false when both ~p and q are false together.

The last column is the entire expression ~q ^ (~p v q). We apply a conjunction to the ~q column and the (~p v q) column. Both expressions must be true for the entire ~q ^ (~p v q) to be true, otherwise it's false.

Let me know if you have any questions. It's probably tricky to wrap your head around at first, but hopefully the table clears things up.

The scratch work section is to show how the fifth and sixth columns are formed.