3.4 and 3.5 Truth Tables for Conditional and Biconditional Statements & Equivalent Statements

Conditional Statements (If-Then 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:

If-Then statements are ONLY false when the IF-PART is TRUE and the THEN-PART is false.  If the IF-PART is not true, then the THEN-PART does not have to follow.
 

The Converse

The "Converse" of PQ  is QP.  The truth table for Q  →  P  compared to PQ 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 If-Part with the Then-Part in a conditional to give us an equivalent statement for all cases.

The Inverse

The "Inverse" of PQ  is ~P~Q.  The truth table for ~P → ~Q  compared to PQ 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 If-Part and the Then-Part 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 PQ  is ~Q~P.  The truth table for ~Q → ~P  compared to PQ 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 If-Part and the Then-Part 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 then-parts 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 (If-and-only-If 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:

If-and-only-if 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 if-and-only-if statements:

P If-and-only-if 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 If-then statement is only false if the IF-Part is true yet the THEN-Part is false.  In the Venn diagram below, this is shown by placing an X within the A circle (IF-Part True) that lies outside the B circle (THEN-Part False). The result would be a false If A, Then B statement.  If, however, the X was within the

If the IF-part is False, the IF-THEN statement is still true. Think of this as a scientific theory that you are trying to disprove. The IF-Part 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 Then-part), does not follow.  For example, if someone theorized, "If a 12-gauge 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 12-gauge 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 IF-AND-ONLY-IF 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 If-and-only-if 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 real-life 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 Vita-max 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 if-then 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.