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# Truth Tables and Logical Statements

Q: How do I make truth tables?!?

A:

Statement 1: (not p) or (q)

OK… Here is how we start.  Image that “p” and “q” are statements that can either be true or false.  I am going to fill in the table with all possible combinations:  p can be true & q can be false, p can be false & q can be true, they can both be true, or they can both be false:

 p q (not p) or (q) T F F T T T F F

OK… all truth tables will look like this to start… Now, the “hard” part is filling in the last column. In this question, we are looking for combinations that are (not p) or (q). So, this means that either “p is false” or “q is true”… Any row that has an F under p or any row that has a T under q will be T.  If not, it will be F.

 p q (not p) or (q) T F F F T T T T T F F T

Statement 2: If (not p) then (q)

The if…then statements, in my opinion, are a little more difficult for truth tables.  We can still do it though!

RULE TO MEMORIZE: “a implies b” is the same as “not a or b”

So, “not p implies q” is the same as “not (not p) or q” = “p or q”…. This means to put a T if either p is T or q is T:

Put in the truth table and fill it out:

 p q p or q T F T F T T T T T F F F

Statement 3: If (p or not q) then (p and not q)

OK… This is the same as:  ” not (p or not q) or (p and not q) ”

HERE IS ANOTHER RULE  not (p or q) = not p and not q.  If you want to distribute the “not” through the parentheses, you must switch the or to and (or switch the and to or)… Also remember, a “not not” makes a positive!  So, back to our statement:

” not (p or not q) or (p and not q) ” = “not p and q or p and not q”

I am going to add our two options to the table along with the combination:

 p q (not p and q) (p and not q) (not p and q) or (p and not q) T F F T T T F F

So, now I am going to fill in column 3, 4 and 5:

Column 3:  not p and q

Put a T if:  q = F and p = T

Column 4:  p and not q

Put a T if:  p = T and q = F

Column 5: Put a T if:  Either column 3 or column 4 is T.

Fill it in like so:

 p q (not p and q) (p and not q) (not p and q) or (p and not q) T F F T T F T T F T T T F F F F F F F F

Statement 4: If (p and q) then (p or q)

So, convert like before:

If (p and q) then (p or q) = not (p and q) or (p or q) = not p or not q or p or q

 p q (not p or not q) (p or q) (not p or not q) or (p or q) T F F T T T F F

Column 3:  not p or not q

Put T if:  p = F or q = F

Column 4:  p or q

Put T if:  p = T or q = T

Column 5:  (not p or not q) or (p or q)

Put a T if either column 3 = T or column 4 = T

Fill in like so:

 p q (not p or not q) (p or q) (not p or not q) or (p or q) T F T T T F T T T T T T F T T F F T F T