Survey of Math - MAT 140
Page: 1
Truth Tables for Negation, Conjunction, and
Disjunction
You may want to purchase Graph paper, (centimeter or quarter inch) and highlight markers. You could also use color
pencils, but you have to be well organized before you can use them.
Here we are going to concern ourselves with the Truth Value of simple statements.
Whether they are True ( T ) or False ( F )
Lets look at the simplest of the truth tables:
1
NOT Truth Table
Here there are only two possibilities for any individual statement: True or False, if we negate these we just get their
opposites.
p ~p
T
F
F
T
2
Truth Table Set up For Two Simple Statements
To account for all the possible combinations of Truths between two simple statements we use the General Setup For
Two Statements will have 4 possibilities:
p q
T T
T F
F T
F F
3
Truth Table for AND
To see how the word AND works with simple statements, imagine that you promise your signi cant other that: "
Honey I am going to the store and to the bank. " . First we will state the simple statements in this sentence:
p : I am going to the store.
q: I am going to the bank.
Thus the sentence symbolically will be : p ^ q.
Now a T will represent that you actually did what you said you would . An F will represent that you did Not do what
you said
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 2
(you lied!).
* So in the TT row you did both of the things that you said you were going to do, thus you told the Truth: Value=T
* In the TF row, you went to the store, but you didn't go to the bank. But you said you would do both so: Value=F
* In the FT row , you didn't go to the store, but you did go to the bank. But again you said you would do both: Value=F
*In the FF row, you told an outright lie!!! Value=F
p
T
T
F
F
q
T
F
T
F
p
^
T
F
F
F
q
There is an easy English explanation of the above: "Only True when Both are True"
Example 1 Construct a Truth Table for: ~ p ^ q
Before we start this, let me explain a little about construction of a truth table. In the table the rst columns that are
lled in will be the ~ p and the q: From here we will write the Answer to the table between these letters.
Notice in the rst row that p D T and q D T: and the statement is ~ p ^ q: With a little substitution we get the rst
line to read: ~ .T / ^ T DF ^ TD F: Thus F is put between the F and the T This step is done in your head and is
not written down. The way that I approach this problem is to "search" for the row(s) where there are two trues and
mark it T since we know that with an AND statement that : "Only True when Both are True". This happens in the
third row, and the rest must be false.
then
Thus the answer is ~ p ^ q D F F T F
4
Truth Table for OR
Again we want to think about promising our signi cant other that " I am going to the store or the bank " Again the
simple statements will be:
p : I am going to the store.
q: I am going to the bank.
And the symbolic promise will be: p _ q
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 3
Here the promise is a little more open. You only really have to do one of the tasks, or both, to keep your word.
But if you don't do either of them, then you "lied"! So the Truth Table looks like:
p
T
T
F
F
q
T
F
T
F
p
_
T
T
T
F
q
There is an easy English explanation of the above: "Only FALSE when Both are FALSE"
Example 2 Construct the truth table for .~ p _ q/ ^ ~q
Here we are going to treat the parentheses as an individual problem that we need to nd the answer for, thus in the
Truth Table we will start by lling in the p's and q's under .~ p _ q/ : Look for the two F 0 s and put the answer of F
between them. If there are no others, the rest must be T
combine this ans. with .~q/ using ^
The last column of T's and F's is the complete answer for this problem. Notice that after we combine two columns
that they are irrelevant for the rest of the problem, above I left them off, but in your work they remain there, you just
have to ignore them. This is the main reason to use a color for 2 combined columns so you know which ones you have
already used!
Thus: .~ p _ q/ ^ ~q D F F F T
A trickier truth table is on in which the Negation .~/ is outside a grouped statement. For example:
~ . p _ ~q/
We must be careful that we do not associate the Negation with a single statement because it is for the Whole group!
Example 3 Construct the Truth Table for: ~ . p _ ~q/ . Now here it is important that you do not associate the outside
~ with the p on the inside.
It will be the last thing that we take care of.
So we will start with the expression inside the parentheses.
Then we need to negate the inside answer
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 4
Thus the answer: ~ . p _ ~q/ D F F T F
5
Set up for Three Simple Statements
With three statements, that have two possible values each, there will be 8 different combinations of T's and F's.
23 D 8: To start, cut the 8 in half, and give 4 T's and 4 F's to the p. For the q, cut the 4 in half and give in order 2
T's and 2 F's. For the r; just alternate between T and F...
Example 4 Find Truth Table for: . p ^ ~q/ _ r
Again we will start with the grouped statement, nd this answer, and put this answer with what is left:
the next operation will be _
6
Truth Value
Now there is another similar type of problem; called Truth Value of a statement. There is no need to do a Truth Table
for these. We just need to substitute the given Truth Values of the Simple statements and Evaluate the expression.
Example 5 Give that p is True q is False and r is False, nd the truth value of: ~ . p _ ~q/ ^ ~r
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 5
To do this, we will use Strict Substitution: The rst step is to replace each p,q, and r with an empty set of ./.
~ .~ p _ q/ ^ ~r
~ [~ ./ _ ./] ^ ~ ./
Then into these empty ./ we insert the Values of each: p is True q is False and r is False
~ [~ .T / _ .F/] ^ ~ .F/
Then simplify each simple statement, starting with the negations and then groups...
~ [F _ F] ^ T
~ [F] ^ T
T ^T
T
Thus the statement ~ . p _ ~q/ ^ ~r with the conditions of p is True q is False and r is False is a TRUE statement...
Example 6 Find the Truth Value of the Following Statement:
" Alaska is not an island, and Hawaii is in the Atlantic or NC does not have a desert."
First step would be to put the Simple statements in symbolic form. Remember that simple statements do Not have the
word NOT in them!!!
p: Alaska is an island . f alse/
q: Hawaii is in the Atlantic . f alse/
r: NC does have a desert . f alse/
Symbolically the statement would be:
~ p ^ .q _ ~r /
~ ./ ^ [./ _ ~ ./]
~ .F/ ^ [.F/ _ ~ .F/]
T ^ [F _ T ]
T ^T
T
Thus the statement is True! It is not necessary that all the pieces be true for a true statement. That's how laws and
lawyers work!!!
Copyright 2007 by Tom Killoran