3.4 and 3.5 Truth Tables for Conditional and
Biconditional Statements & Equivalent Statements
Conditional Statements (IfThen Statements)
The truth table for P →
Q is shown below.
P 
Q 
P → Q 
T 
T 
T 
T 
F 
F 
F 
T 
T 
F 
F 
T 
Here's all you have to remember:
IfThen statements are ONLY false when the IFPART is TRUE and
the THENPART is false. If the IFPART is not true, then the
THENPART does not have to follow.
The Converse
The "Converse" of P→Q is Q→P.
The truth table for Q → P
compared to P→Q is shown below.
P 
Q 
Q → P 
P → Q 
T 
T 
T 
T 
T 
F 
T 
F 
F 
T 
F 
T 
F 
F 
T 
T 
Note: The Converse is NOT logically equivalent to the original
conditional. We can never switch the IfPart with the ThenPart in a
conditional to give us an equivalent statement for all cases.
The Inverse
The "Inverse" of P→Q is ~P→~Q.
The truth table for ~P → ~Q
compared to P→Q is shown below.
P 
Q 
~P 
~Q 
~P → ~Q 
P → Q 
T 
T 
F 
F 
T 
T 
T 
F 
F 
T 
T 
F 
F 
T 
T 
F 
F 
T 
F 
F 
T 
T 
T 
T 
Note: The Inverse is NOT logically equivalent to the original
conditional. We can never take the negations of both the IfPart and
the ThenPart in a conditional to give us an equivalent statement for all
cases. Note however that the converse is logically equivalent to the
inverse.
The Contrapositive
The "Contrapositive" of P→Q
is ~Q→~P. The truth table for ~Q →
~P compared to P→Q is
shown below.
P 
Q 
~P 
~Q 
~Q → ~P 
P → Q 
T 
T 
F 
F 
T 
T 
T 
F 
F 
T 
F 
F 
F 
T 
T 
F 
T 
T 
F 
F 
T 
T 
T 
T 
Note: The Contrapositve IS logically equivalent to the original
conditional. We can always take the negations of both the
IfPart and the ThenPart in a conditional and switch them with each other
to give us an equivalent statement for all cases. This is the reason that
the converse is logically equivalent to the inverse  the if and
thenparts are switched and negations are taken.
Example: Write the converse, inverse, and contrapositive of the
statement: "If it rains on me, then I get wet."
Converse: "If I get wet, then it has rained on me".
Inverse: "If it does not rain on me, then I don't get wet".
Contrapositive: If I do not get wet, then it did not rain on me.
Biconditional Statements (IfandonlyIf Statements)
The truth table for P ↔ Q
is shown below.
P 
Q 
P → Q 
T 
T 
T 
T 
F 
F 
F 
T 
F 
F 
F 
T 
Here's all you have to remember:
Ifandonlyif statements are ONLY true when P and Q are BOTH
TRUE or when P and Q are BOTH FALSE.
Alternative Truth Table For P ↔ Q
Since P ↔ Q is logically
equivalent to
(P→Q)Λ(Q→ P). The truth table for
P ↔ Q written in this way is
shown below.
P 
Q 
P → Q 
Q → P 
(P → Q)
Λ (Q
→ P) 
T 
T 
T 
T 
T 
T 
F 
F 
T 
F 
F 
T 
T 
F 
F 
F 
F 
T 
T 
T 
Here's another way to remember ifandonlyif statements:
P Ifandonlyif Q means that events P and Q either both
happen or both don't happen.
Example: x = 2 if and only if x+3 = 5 would only be false if
one statement were true and the other false  something that is
impossible in mathematics as we know it. For all practical
purposes, only 2 of the 4 rows of the if and only if table shown above
are possible for this mathematics example.
We say that ~P V ~Q is logically equivalent to ~(P Λ
Q).
Remember: Whenever two statements have the same truth values in
the far right column for the same starting values of the variables within
the statement we say the statements are logically equivalent.
In Summary:
An Ifthen statement is only false if the IFPart is true yet the
THENPart is false. In the Venn diagram below, this is shown by
placing an X within the A circle (IFPart True) that lies outside the B
circle (THENPart False). The result would be a false If A, Then B
statement. If, however, the X was within the
If the IFpart is False, the IFTHEN statement is still true.
Think of this as a scientific theory that you are trying to disprove. The
IFPart is the hypotheses of the theory. To disprove the theory, you
must first satisfy the hypothesis and then find a case where the
conclusion (the Thenpart), does not follow. For example, if someone
theorized, "If a 12gauge wire is made of plastic, then it can be used to
transmit electricity as well as a copper wire", you would disprove this
theory by constructing a 12gauge wire made of plastic and then showing it
does not conduct electricity as well as a copper wire of the same gauge.
You could not disprove the theory by testing a wire made of aluminum! The
figure below illustrates this for IF A, THEN B.
A IFANDONLYIF B really means A and B represent the same events or
events that occur (or don't occur) simultaneously. Using the Venn
Diagram, this would mean A and B represent the same circle as shown below.
A Ifandonlyif B statements are true if BOTH A & B parts are true or
BOTH A & B parts are false.
De Morgan's Laws
De Morgan's Laws are two special cases of logical equivalence. De
Morgan's Laws state:
~P V ~Q is logically equivalent to
~(P Λ Q)
~P Λ ~Q
is logically equivalent to ~(P V Q)
You may show these equivalent relationships with truth tables.
Example: Find the negation of (P V ~Q) and also the negation of
(~P ^ ~Q).
Solution: DeMorgan's Laws basically say that if you negate an OR or AND
statement, you negate each component and switch the OR to AND or switch
AND to OR.
~(P V ~Q) ≡ ~P ^ Q
~(~P ^ ~Q) ≡ P V Q
Assignment
1. Complete the truth table below. Note that
(P=>Q)^(Q=>P) is the logical equivalent of P if and only if Q.
P 
Q 
P=>Q 
Q=>P 
(P=>Q) ^ (Q=>P) 
T 
T 



T 
F 



F 
T 



F 
F 



2. Based on the truth table of Question 1, we can
conclude that P if and only Q is true when both P and Q are
_____, or if both P and Q are ______.
3. Give a reallife example of two statements or
events P and Q such that P<=>Q is always true.
4. Make a truth table for ~(~P ^ Q) and also one
for PV~Q. Based on the table(s), can we conclude ~(~P ^ Q) is logically
equivalent to PV~Q? Include the truth tables with your answer.
5. Use DeMorgan's Laws to write the negation of
"I am cold and I am not happy". Your answer should be an OR statement.
6. Write the converse, inverse, and contrapositve
of "If I work hard, then I have money." Is the contrapositive (~Q=>~P)
always logically equivalent to the original statement (P=>Q)? Is the
converse (Q=>P) always logically equivalent to the inverse (~P=>~Q)?
7. How do you go about proving that P=>Q is a
false statement? In other words, what truth values of P and Q are needed?
8. An advertisement states, "If you take Vitamax
Ultra Vitamins 12 times per day at 1 hour intervals for 90 days straight,
then you will feel better." Why would it be difficult to prove that
this ifthen statement is false?
9. Use a truth table or tables to show that
~(P=>Q) is logically equivalent to P ^ ~Q.
10. Based on the results of Question 9, write the
negation of "If I take aspirin, then I will feel better" as an AND
statement.

IMPORTANT NOTE: When you name your file, do NOT use
& or any other character like
$,@,#, etc. in the file name  it will make the file unreadable for
me. For example, a file named 3.4&3.5 will NOT open
for me! Instead, name it 3_4and3_5 or 3_45. 
