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.
Show Solution
The first step to creating a truth table for any compound statement is always to list all the possible truth value combinations for the simple statements, in this case, P, Q, and R. As we have noticed before, since there are two possible truth values for each of the three simple statements, then there will be [latex]2\cdot2\cdot2=8[/latex] combinations. We write them in pattern: the first column contains 4 Ts followed by 4 Fs, the second column contains 2 Ts, 2 Fs, then repeats, and the last column alternates. This pattern ensures that all combinations are considered.
| [latex]P[/latex] |
[latex]Q[/latex] |
[latex]R[/latex] |
| T |
T |
T |
| T |
T |
F |
| T |
F |
T |
| T |
F |
F |
| F |
T |
T |
| F |
T |
F |
| F |
F |
T |
| F |
F |
F |
Part 2: Work from the inside out.
Show Solution
It helps to work from the inside out when creating truth tables, and create tables for intermediate operations. In the following table, we list the truth values for the innermost expression, [latex]Q\vee{R}[/latex].
| [latex]P[/latex] |
[latex]Q[/latex] |
[latex]R[/latex] |
[latex]Q\vee{R}[/latex] |
| T |
T |
T |
T |
| T |
T |
F |
T |
| T |
F |
T |
T |
| T |
F |
F |
F |
| F |
T |
T |
T |
| F |
T |
F |
T |
| F |
F |
T |
T |
| F |
F |
F |
F |
Part 3: Move out to the next layer.
Show Solution
Next we can find the negation of [latex]Q\vee{R}[/latex], working off the [latex]Q\vee{R}[/latex] column we just created.
| [latex]P[/latex] |
[latex]Q[/latex] |
[latex]R[/latex] |
[latex]Q\vee{R}[/latex] |
[latex]\sim\left(Q\vee{R}\right)[/latex] |
| T |
T |
T |
T |
F |
| T |
T |
F |
T |
F |
| T |
F |
T |
T |
F |
| T |
F |
F |
F |
T |
| F |
T |
T |
T |
F |
| F |
T |
F |
T |
F |
| F |
F |
T |
T |
F |
| F |
F |
F |
F |
T |
Part 4: Put it all together.
Show Solution
Finally, we find the values of [latex]P[/latex] and [latex]\sim\left(Q\vee{R}\right)[/latex]
| [latex]P[/latex] |
[latex]Q[/latex] |
[latex]R[/latex] |
[latex]Q\vee{R}[/latex] |
[latex]\sim\left(Q\vee{R}\right)[/latex] |
[latex]P\,\wedge\sim\left(Q\,{\vee}R\right)[/latex] |
| T |
T |
T |
T |
F |
F |
| T |
T |
F |
T |
F |
F |
| T |
F |
T |
T |
F |
F |
| T |
F |
F |
F |
T |
T |
| F |
T |
T |
T |
F |
F |
| F |
T |
F |
T |
F |
F |
| F |
F |
T |
T |
F |
F |
| F |
F |
F |
F |
T |
F |
It turns out that this complex expression is only true in one case: if [latex]P[/latex] is true, [latex]Q[/latex] is false, and [latex]R[/latex] is false.
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?
Show Solution
It is a true statement because if the antecedent “it is raining” is true, then the consequence “there are clouds in the sky” must also be true.
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.
Show Solution
There are four possible outcomes:
- You upload the picture and keep your job.
- You upload the picture and lose your job.
- You don’t upload the picture and keep your job.
- You don’t upload the picture and lose your job.
There is only one possible case where your friend was lying—the first option where you upload the picture and keep your job. In the last two cases, your friend didn’t say anything about what would happen if you didn’t upload the picture, so you can’t conclude their statement is invalid, even if you didn’t upload the picture and still lost your job.
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
Show Solution
As usual, we start by listing all the possible truth value combinations for the simple statements. As in our last case, we have three simple statements. Thus, we will again have 8 different possible combinations of truth values.
| [latex]p[/latex] |
[latex]q[/latex] |
[latex]r[/latex] |
| T |
T |
T |
| T |
T |
F |
| T |
F |
T |
| T |
F |
F |
| F |
T |
T |
| F |
T |
F |
| F |
F |
T |
| F |
F |
F |
Part 2: Make truth table for the antecedent.
Show Solution
Next, we construct the following truth table for the antecedent. To do this, we will also need the negation of [latex]q[/latex] so we add a column for that.
| [latex]p[/latex] |
[latex]q[/latex] |
[latex]r[/latex] |
[latex]\sim{q}[/latex] |
[latex]p\,\wedge\sim{q}[/latex] |
| T |
T |
T |
F |
F |
| T |
T |
F |
F |
F |
| T |
F |
T |
T |
T |
| T |
F |
F |
T |
T |
| F |
T |
T |
F |
F |
| F |
T |
F |
F |
F |
| F |
F |
T |
T |
F |
| F |
F |
F |
T |
F |
Part 3: Put it all together.
Show Solution
Finally, we build the truth table for the implication. Notice that we moved the column containing the truth values for [latex]r[/latex] to make it easier to find the truth values for the implication.
| [latex]p[/latex] |
[latex]q[/latex] |
[latex]\sim{q}[/latex] |
[latex]p\,\wedge\sim{q}[/latex] |
[latex]r[/latex] |
[latex]\left(p\,\wedge\sim{q}\right)\rightarrow{r}[/latex] |
| T |
T |
F |
F |
T |
T |
| T |
T |
F |
F |
F |
T |
| T |
F |
T |
T |
T |
T |
| T |
F |
T |
T |
F |
F |
| F |
T |
F |
F |
T |
T |
| F |
T |
F |
F |
F |
T |
| F |
F |
T |
F |
T |
T |
| F |
F |
T |
F |
F |
T |
In this case, when [latex]p[/latex] is true, [latex]q[/latex] is false, and [latex]r[/latex] is false, then the antecedent [latex]p\,\wedge\sim{q}[/latex] will be true but the consequence false, resulting in a invalid implication; every other case gives a valid implication.
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.
Show Solution
Converse: “If there are clouds in the sky, it is raining.” This is certainly not always true.
Contrapositive: “If there are not clouds in the sky, then it is not raining.” This statement is valid.
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.
