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Discrete Mathematics: An Active Approach to Mathematical Reasoning

Jennifer Firkins Nordstrom

Section 3.4 Arguments with Quantifiers

In Section 2.3 we studied arguments with logical statements. In this chapter we’ve looked at statements with quantifiers. Now we want to combine the two ideas.
Recall, an argument is valid if it whenever the premises are true, the conclusion must be true. An argument is invalid if it possible for the premises to be true and the conclusion false.

Example 3.4.1. An Argument with Quantified Statements.

All computer science majors take Discrete Math before graduation.
Anna is a computer science major.
\(\therefore\ \)Anna takes Discrete Math before graduation.
Is this argument valid or invalid?
Answer.
This argument is valid.

Example 3.4.2. Another Argument with Quantified Statements.

All computer science majors take Discrete Math before graduation.
Troy did not take Discrete Math before graduation.
\(\therefore\ \)Troy is not a computer science major.
Is this argument valid or invalid?
Answer.
This argument is valid.
Although Example 3.4.1 and Example 3.4.2 are a valid arguments, we will have trouble using a truth-table to determine validity. In this section we will develop other tools for analyzing quantified arguments. These examples are extentions of modus ponens and modus tollens from Some Common Forms for Valid Arguments.

Common Forms of Arguments with Quantifiers.

Universal modus ponens:
\(\forall x\text{,}\) if \(P(x)\) then \(Q(x)\)
\(P(a)\)
\(\therefore\ Q(a)\)
Example 3.4.1 is an example of universal modus ponens.
Universal modus tollens:
\(\forall x\text{,}\) if \(P(x)\) then \(Q(x)\)
\(\sim Q(a)\)
\(\therefore\ \sim P(a)\)
Example 3.4.2 is an example of universal modus tollens.
To determine validity of quantified arguments we need to ask if it is possible for the premises to be true while the conclusion is false. If it is possible, then the argument is invalid. If it is not possible, then the argument is valid. We will use Venn diagrams to help us determine validity. A Venn diagram is just a picture where we use circles to represent sets of objects and points to represent specific elements. We draw the circles to represent how our sets are related to each other.
The Venn diagram for Example 3.4.1 is given by the following figure.
Figure 3.4.3. Venn diagram for the argument in Example 3.4.1
The picture represents the two premises of the argument. In the Venn diagram we have a circle representing students taking Discrete Math (“Discrete Math”) and a circle representing computer science majors (“CS majors”). The CS major circle is completely inside the Discrete Math circle since the first statement in the argument is that all computer science majors take Discrete Math. This means the set of computer science majors is a subset of the set of students taking Discrete Math. The second statement in the argument is that Anna is a computer science major. This means Anna is an element (represented by a point) of the set of CS majors.
Now, if we look at the diagram determined by the two premises, we see that Anna must also be an element of the set of students taking Discrete Math. Thus, it is not possible to have true premises and a false conclusion.

Process for Using a Venn diagram to Determine Validity.

  1. Draw the Venn diagram for each premise.
    Statements such as “All A are B” mean A is a subset of B.
  2. Make sure you distinguish between sets (lots of things have a certain property) and elements (a specific thing with a property).
  3. See if it is possible for the premises to be true and the conclusion false.
    Sometimes there is a choice about where to place a certain element of set. For example, if I know \(x\) is not in set A, then \(x\) might be in set B or it might not be in B. We can use these choices to see if we can make the premises true and the conclusion false.
Let’s use this process on some more examples.

Example 3.4.4. Determining Validity.

All integers are rational numbers.
\(\sqrt{2}\) is not rational.
\(\therefore\ \sqrt{2}\) is not an integer.
The two premises are represented with the following diagram. Recall, \(\mathbb{Q}\) is the set of rational numbers and \(\mathbb{Z}\) is the set of integers.
We can see from the diagram that it is impossible to make the conclusion false. In particular, \(\sqrt{2}\) cannot be in the set of integers.

Activity 3.4.1.

Consider the general form of universal modus ponens.

(a)

Draw the Venn diagram for universal modus ponens. You should have a circle labeled “P” and one labeled “Q.”

(b)

Write your own argument that has the form of universal modus ponens.

(c)

Use a Venn diagram with appropriate labels to show that your argument is valid.

Activity 3.4.2.

Consider the general form of universal modus tollens.

(a)

Draw the Venn diagram for universal modus tollens. You should have a circle labeled “P” and one labeled “Q.”

(b)

Write your own argument that has the form of universal modus tollens.

(c)

Use a Venn diagram with appropriate labels to show that your argument is valid.

Example 3.4.5. An Invalid Argument.

All integers are rational numbers.
\(x\) is rational.
\(\therefore\ x\) is an integer.
The first premise is represented with the following diagram.
But we need to be careful about where we place \(x\text{.}\) The second premise just says it needs to be inside the rational circle. Is it possible to place it in the rational circle so that the conclusion is false? Yes, we can place \(x\) inside the rational circle, but not in the integer circle. So the following diagram shows how it is possible to have true premises and a false conclusion.
Therefore, the argument is invalid.

Invalid Arguments with Quantifiers.

Converse error
\(\forall x\) if \(P(x)\) then \(Q(x)\)
\(Q(a)\)
\(\therefore\ \)\(P(a)\)
Example 3.4.5 is an example of converse error.
Inverse error
\(\forall x\) if \(P(x)\) then \(Q(x)\)
\(\sim P(a)\)
\(\therefore\ \)\(\sim Q(a)\)

Activity 3.4.3.

Write your own example of inverse error. Use a Venn diagram to show your argument is invalid.
It will be useful to have some standard Venn diagrams for common forms of statements.

Venn Diagrams for Common Statements.

  • All B are A.
  • Some A are B, or there exists an A that is a B.
  • No A are B.

Activity 3.4.4.

For each argument use a Venn diagram to determine if the argument is valid or invalid.

(a)

Every Linfield student lives on campus.
Freya lives on campus.
\(\therefore\) Freya is a Linfield student.

(b)

Every polynomial function is differentiable.
\(f(x)=|x|\) is not a polynomial function.
\(\therefore f(x)=|x|\) is not differentiable.

Activity 3.4.5.

For each argument use a Venn diagram to determine if the argument is valid or invalid.

(a)

\(\forall\ x\text{,}\) if \(x\) studies then \(x\) will do well on the exams.
\(\forall\ x\text{,}\) if \(x\) does well on the exams then \(x\) will get an A in Discrete Math.
\(\therefore\) If Gauss studies then Gauss will get an A in Discrete Math.

(b)

\(\forall\ x\text{,}\) if \(x\) studies then \(x\) will do well on the exams.
\(\forall\ x\text{,}\) if \(x\) does well on the exams then \(x\) will get an A in Discrete Math.
\(\therefore\) Newton will get an A in Discrete Math.

(c)

Compare your answers to the two arguments. What is different? How does the difference affect the argument?

Reading Questions Check Your Understanding

1.

    Determine whether the following argument is valid or invalid.
    All zebras have stripes.
    Rex has stripes.
    \(\therefore\ \) Rex is a zebra.
  • The argument is valid.
  • The argument is invalid.

2.

    Determine whether the following argument is valid or invalid.
    All zebras have stripes.
    Lily does not have stripes.
    \(\therefore\ \) Lily is not a zebra.
  • The argument is valid.
  • The argument is invalid.

3.

    Determine whether the following argument is valid or invalid.
    All zebras have stripes.
    All zebras are animals.
    Rex has stripes.
    \(\therefore\ \) Rex is an animal.
  • The argument is valid.
  • The argument is invalid.

4.

    Determine whether the following argument is valid or invalid.
    All zebras have stripes.
    All animals have stripes.
    \(\therefore\ \) All zebras are animals.
  • The argument is valid.
  • The argument is invalid.

5.

    Determine whether the following argument is valid or invalid.
    All zebras have stripes.
    All things with stripes are animals.
    \(\therefore\ \) All zebras are animals.
  • The argument is valid.
  • The argument is invalid.

6.

    Determine whether the following argument is valid or invalid.
    All zebras are animals.
    All animals have stripes.
    \(\therefore\ \) All zebras have stripes.
  • The argument is valid.
  • The argument is invalid.

Exercises Exercises

1.

Determine if the argument is valid or invalid. If it is valid, state whether it is an example of modus ponens or modus tollens. If it is invalid, state whether it exhibits converse error or inverse error.
  1. All healthy people eat an apple a day.
    Keisha eats an apple a day.
    \(\therefore\) Keisha is a healthy person.
  2. If the product of two numbers is 0, then at least one of the numbers is 0.
    For a particular number \(x\text{,}\) neither \((2x+1)\) nor \((x-7)\) equals 0.
    \(\therefore\) The product \((2x+1)(x-7)\) is not 0.
  3. All honest people pay their taxes.
    Darth is not honest.
    \(\therefore\) Darth does not pay his taxes.
  4. Any sum of two rational numbers is rational.
    The sum \(r+s\) is rational.
    \(\therefore\) The numbers \(r\) and \(s\) are both rational.
  5. If an infinite series converges, then its terms go to 0.
    The terms of the infinite series \(\sum_{n=1}^{\infty}\frac{1}{n}\) go to 0.
    \(\therefore\) The infinite series \(\sum_{n=1}^{\infty}\frac{1}{n}\) converges.
  6. If an infinite series converges, then its terms go to 0.
    The terms of the infinite series \(\sum_{n=1}^{\infty}\frac{n}{n+1}\) do not go to 0.
    \(\therefore\) The infinite series \(\sum_{n=1}^{\infty}\frac{n}{n+1}\) does not converge.

2.

Determine if the argument is valid or invalid. Support your answer by drawing a Venn diagram.
  1. All people are mice.
    All mice are mortal.
    \(\therefore\) All people are mortal.
  2. All polynomial functions are differentiable.
    All differentiable functions are continuous.
    \(\therefore\) All polynomial functions are continuous.

3.

Use a diagram to show the following argument can have true premises and a false conclusion.
All dogs are carnivorous.
Aaron is not a dog.
\(\therefore\) Aaron is not carnivorous.

4.

What can you conclude about the validity or invalidity of the following argument form? Explain your answer.
\(\forall x\text{,}\) if \(P(x)\) then \(Q(x)\text{.}\)
\(\forall x\text{,}\) if \(Q(x)\) then \(R(x)\text{.}\)
\(\therefore\) \(P(a)\) and \(R(a)\text{.}\)

5.

What can you conclude about the validity or invalidity of the following argument form? Explain your answer.
\(\forall x\text{,}\) if \(P(x)\) then \(Q(x)\text{.}\)
\(\forall x\text{,}\) if \(Q(x)\) then \(R(x)\text{.}\)
\(\therefore\) \(P(a)\rightarrow R(a)\text{.}\)
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