BCCC ASC Rev. 6/2019
Truth Tables
Truth tables are used to determine the validity or truth of a compound statement*.
• A compound statement is composed of one or more simple statements. Simple statements are
typically represented by symbols (often letters).
• Each symbol represents a statement such as “John scored a goal” or “It is raining.”
• Constructing a truth table for a compound statement depends upon the simple statements
composing the compound statement
*The term statement may also be referred to as a premise or expression depending on the context.
To construct a truth table for a compound statement that consists of two simple statements,
begin by listing the four true-false cases shown below:
To construct a truth table for a compound statement that consists of three simple statements,
begin by listing the eight true-false cases shown below:
p
q
r
The number of times T appears consecutively in each column is determined by
the number of statements.
In this example, the formula for the number of times T is listed consecutively
under statement p is 2
2
, the formula for the number of times T is listed
consecutively under statement q is 2
1
, and the formula for the number of times
T is listed consecutively under statement r is 2
0
.
One good way to remember the sequence is to remember that the number of
times T is listed consecutively in the first statement is half the number of true-
false cases. For the next statement, it would be half of the first, and so on.
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
Note that the number of true-false cases has doubled with the addition of one statement. The
total number of cases will be determined by using 2 to the power of the number of statements.
This means that if there are four statements, we would determine the number of cases by
calculating 2
4
.
p
q
If we use the
example
statements above,
we can see what
this would translate
to.
John scored a goal.
In the first row,
John scored a goal,
and it is raining are
true. Both events
occurred.
T
T
T
T
F
T
F
T
F
F
F
F
BCCC ASC Rev. 6/2019
Connectives
The validity or truth of the compound statement is dependent upon the simple statements and the connective used.
Connectives are symbols that indicate the relationship between simple statements.
The five most common connectives are listed below.
If multiple connectives are used, the truth of the connectives must be completed in a particular order (see
below for details).
In the following examples, p represents the statement “John scored a goal” and q represents the statement “John
won the game.”
Negation - “ Not “ - Complete First
p
~p
The statement is true when the input statement is false.
The statement is false when the input statement is true.
~p represents the statement “John did not score a goal.”
T
F
F
T
Conjunction – “and” - Complete Second along with Disjunction
p
q
p Λ q
The statement is true only when both input statements are true.
Otherwise, the statement is false.
p ^ q represents the statement “John scored a goal and won the game.”
T
T
T
T
F
F
F
T
F
F
F
F
Disjunction – “or” - Complete Second along with Conjuction
p
q
p V q
The statement is false only when both input statements are false.
Otherwise, the statement is true.
p V q represents the statement “John scored a goal or won the game.”
T
T
T
T
F
T
F
T
T
F
F
F
Conditional – “if ….. then” - Complete Third
p
q
p → q
The statement is false only when the first input statement is true,
and the second input statement is false.
Otherwise, the statement is true.
p→q represents the statement “If John scored a goal, then won the
game.”
T
T
T
T
F
F
F
T
T
F
F
T
Biconditional – “if and only if” - Complete Last
P
q
p ↔ q
The statement is true when both input statements are both true,
or both false.
Otherwise, the statement is false.
p ↔ q represents the statement “John scored a goal if and only if he
won the game.”
T
T
T
T
F
F
F
T
F
F
F
T