Truth Tables
MATH 102
Contemporary Math
S. Rook
Overview
• Section 3.2 in the textbook:
– Listing all truth possibilities
– Truth tables for negation & connectives
– Evaluating compound statements
– Logically equivalent statements
Listing All Truth Possibilities
Listing All Truth Possibilities
• Consider a statement p
– What are the only possible truth values of p?
• Consider statements p and q
– We know the possible truth values of p and q
separately
– What are all the possible truth values if we pick a
value for p and then pick a value for q?
• What are all the possible truth values for
statements p, q, and r?
• Notice the pattern?
Listing All Truth Possibilities
(Continued)
• There will be 2k total possibilities for k
statements
– Need to account for all possibilities
– Use the half-and-half method to capture all
possibilities
– Populating the truth table with all possible truth
values of the variables is very important to
master!
• You must be able to work with a truth table containing
up to 3 variables
Truth Tables for Negation &
Connectives
Truth Tables
• Truth table: systematic way of determining the truth
values of a compound statement by examining all the
possible truth values of its input statements
– We looked extensively on populating a truth table with
input values
• The goal for now is to be able to construct a truth table
to show the possible truth values for ANY compound
statement
– Later, we will see how to use truth tables to make logical
inferences
• Need to first understand the truth tables for negation
and connectives
Truth Table for Negation
• Consider the statement 2 + 1 = 3
– Is the statement true or false? What about its
negation?
• Consider the statement 8 – 5 = 2
– Is the statement true or false? What about its
negation?
• Negation: If a statement is true, then its negation
is false; if a statement is false, then its negation is
true
p
~p
T
F
F
T
Truth Table for Conjunction
• Let p be the statement I need to buy bread
and q be the statement I need to buy milk
– What is the statement p  q in English?
– What does it mean for p  q to be true?
• Conjunction: True when ALL variables are
true; false otherwise
p
q
pq
T
T
T
T
F
F
F
T
F
F
F
F
Truth Table for Disjunction
• Let p be the statement I washed the cat and q
be the statement I put my shoes on
– What is the statement p v q in English?
– What does it mean for p v q to be true?
• Disjunction: True when at least one variable
is true; false when all variables are false
p
q
pvq
T
T
T
T
F
T
F
T
T
F
F
F
Inclusive Or vs Exclusive Or
• The disjunction defined on the last slide is the
inclusive or
– Version used in logic
• Different from the normally used exclusive or
– i.e. One or the other, but NOT BOTH
– e.g. If a waitress asks you whether you want Coke
or Sprite, what does she expect you to say?
• When you see the word or in this chapter, we
are referring to the inclusive or
Truth Tables (Example)
Ex 1: Create a truth table for the following
statements:
a) p  q  r
b) p  q  r
Evaluating Compound
Statements
Evaluating Compound Statements
• Consider evaluating the compound statement
~  p  q  ~ q
• How many variables are there?
– We know how to construct all possible inputs for a
truth table
• Recall the Order of Operations in Algebra
– Similar construct in logic:
• Parentheses
• Negation
• Conjunction & Disjunction
Evaluating Compound Statements
(Continued)
• Evaluate one negation or one connective per
column
– What should we evaluate first in the example?
• Take ONE STEP at a time and keep adding
columns to the truth table until you arrive at the
desired statement
– Reduces the number of columns under consideration
in each step to 2 or even 1 which is much easier!
• Requires practice in order to master!
Evaluating Compound Statements
(Example)
Ex 2: Construct a truth table:
a) ~  p  q   ~  p  q 
b)
 p ~ q  ~ r
Logically Equivalent Statements
Logically Equivalent Statements
• We say that two statements are logically equivalent
if their truth values match exactly
– Useful to test whether two statements logically mean
the same thing
– Use a truth table
• DeMorgan’s Laws deal with distributing a negation
through parentheses to create a logically equivalent
statement
~  p  q  ~ p  ~ q
~  p  q  ~ p  ~ q
Logically Equivalent Statements
(Example)
Ex 3: Determine whether the pairs of
statements are logically equivalent:
a) ~ ~ p ~ q  ; p  q
b) p  ~ q  ~ r  ;  p  ~ q   p  ~ r 
Summary
• After studying these slides, you should know how to do the
following:
– Populate a truth table with all combinations of truth values
of the inputs
– Know the truth tables for negation, conjunction, and
disjunction
– Evaluate a compound statement using a truth table
– Determine whether pairs of statements are logically
equivalent
• Additional Practice:
– See the list of suggested problems for 3.2
• Next Lesson:
– The Conditional & Biconditional (Section 3.3)