More on Truth Tables

Now that we have created a few truth tables, we will use them to understand more complex statements.  Not surprisingly, mathematicians commonly use symbols for and, or, and not  that can make using a truth table easier.

Symbols

The symbol [latex]\sim[/latex] is used for not: not [latex]P[/latex] is notated [latex]\sim{P}[/latex]

The symbol [latex]\wedge[/latex] is used for and: [latex]P[/latex] and [latex]Q[/latex] is notated [latex]P\wedge{Q}[/latex].

The symbol [latex]\vee[/latex] is used for or: [latex]P[/latex] or [latex]Q[/latex] is notated [latex]P\vee{Q}[/latex]

 

The three truth tables we’ve created so far are reproduced below using these symbols.

Summary of Truth Tables

[latex]P[/latex] [latex]\sim{P}[/latex]
T F
F T
[latex]P[/latex] [latex]Q[/latex] [latex]P\wedge{Q}[/latex]
T T T
T F F
F T F
F F F
[latex]P[/latex] [latex]Q[/latex] [latex]P\vee{Q}[/latex]
T T T
T F T
F T T
F F F

 

Truth tables really become useful when analyzing more complex logical statements.

ExampleS

Create a truth table for the statement [latex]P\wedge\sim\left(Q\vee{R}\right)[/latex]

Part 1: List all possible truth value combinations.

 

Part 2: Work from the inside out.

 

Part 3: Move out to the next layer.

 

Part 4: Put it all together.

Try It

Try It

 

Another important mathematical statement is called an implication.

Implications

An implication is a logical statement suggesting that a phrase p, called the hypothesis or antecedent, implies a consequence q. Implications suggest that the consequence must logically follow if the antecedent is true.

Implications are commonly written as [latex]p\rightarrow{q}[/latex]

Example

The English statement “If it is raining, then there are clouds is the sky” is a logical implication. Is this statement true? Why or why not?

Notice that the statement tells us nothing of what to expect if it is not raining. If the antecedent is false, then the implication becomes irrelevant.

Example

A friend tells you that “if you upload that picture to Facebook, you’ll lose your job.” Describe the possible outcomes related to this statement, and determine whether your friend’s statement is invalid.

In traditional logic, an implication is considered valid (true) as long as there are no cases in which the antecedent is true and the consequence is false. It is important to keep in mind that symbolic logic cannot capture all the intricacies of the English language. Let’s analyze logical implication using a truth table.

Truth Table – Implication

p q p → q
T T T
T F F
F T T
F F T

Example

Construct a truth table for the statement [latex]\left(p\,\wedge\sim{q}\right)\rightarrow{r}[/latex]

Part 1: List all possible truth value combinations

 

Part 2:  Make truth table for the antecedent.

 

Part 3: Put it all together.

Try It

 

For any implication, there are two very important related statements, the converse and the contrapositive.  Let’s investigate each of these statements and their relationships to the original implication.

Statements Related to an implication

Consider the original implication is “if p then q”.

The converse is “if q then p”.

The contrapositive is “if not q then not p”.

Symbolic Representations
Original implication [latex]p\rightarrow{q}[/latex]
Converse [latex]q\rightarrow{p}[/latex]
Contrapositive [latex]\sim{q}\rightarrow\sim{p}[/latex]

Example

Consider again the true implication “If it is raining, then there are clouds in the sky.”

Write the converse and contrapositive statements.

Try It

 

Two statements are considered to be logically equivalent if they have the same truth value for every line of the truth table.  Looking at truth tables, we can see that the original implication and the contrapositive are logically equivalent.  The converse is NOT equivalent to the original implication.

Implication Converse Contrapositive
[latex]p[/latex] [latex]q[/latex] [latex]p\rightarrow{q}[/latex] [latex]q{\rightarrow}p[/latex] [latex]\sim{q}\rightarrow\sim{p}[/latex]
T T T T T
T F F T F
F T T F T
F F T T T

Equivalence of Implications

A conditional statement and its contrapositive are logically equivalent.

The conditional statement and it converse are **NOT** logically equivalent.