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Discrete Mathematics with Applications | 4th Edition | ISBN: 9780495391326 | Authors: Susanna S. Epp
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Some of the arguments are valid by universal

Chapter 3, Problem 14E

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QUESTION:

Problem 14E

Some of the arguments are valid by universal modus ponens or universal modus tollens; others are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers.

Universal Modus Ponens

The rule of universal instantiation can be combined with modus ponens to obtain the valid form of argument called universal modus ponens.

Universal Modus Ponens

Formal Version

Informal Version

∀x, if P(x) then Q(x).

If x makes P(x) true, then x makes Q(x) true.

P(a) for a particular a.

a makes P(x) true.

 ∴ Q(a).

 ∴a makes Q(x) true.

Universal ModusTollens

Another crucially important rule of inference is universal modus tollens.Its validity results from combining universal instantiation with modus tollens. Universal modus tollens is the heart of proof of contradiction, which is one of the most important methods of mathematical argument.

Universal Modus Tollens

Formal Version

Informal Version

∀x, if P(x) then Q(x).

If xmakes P(x) true, then xmakes Q(x) true.

~Q(a), for a particular a

a does not make Q(x) true.

∴ ~P(a).

∴ a does not make P(x) true.

Exercise

If compilation of a computer program produces error messages, then the program is not correct.

Compilation of this program does not produce error messages.

∴ This program is correct.

Questions & Answers

QUESTION:

Problem 14E

Some of the arguments are valid by universal modus ponens or universal modus tollens; others are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers.

Universal Modus Ponens

The rule of universal instantiation can be combined with modus ponens to obtain the valid form of argument called universal modus ponens.

Universal Modus Ponens

Formal Version

Informal Version

∀x, if P(x) then Q(x).

If x makes P(x) true, then x makes Q(x) true.

P(a) for a particular a.

a makes P(x) true.

 ∴ Q(a).

 ∴a makes Q(x) true.

Universal ModusTollens

Another crucially important rule of inference is universal modus tollens.Its validity results from combining universal instantiation with modus tollens. Universal modus tollens is the heart of proof of contradiction, which is one of the most important methods of mathematical argument.

Universal Modus Tollens

Formal Version

Informal Version

∀x, if P(x) then Q(x).

If xmakes P(x) true, then xmakes Q(x) true.

~Q(a), for a particular a

a does not make Q(x) true.

∴ ~P(a).

∴ a does not make P(x) true.

Exercise

If compilation of a computer program produces error messages, then the program is not correct.

Compilation of this program does not produce error messages.

∴ This program is correct.

ANSWER:

Solution:-

Step1

Given that

We have to state that which arguments are valid and which are invalid and Justify answers.

Step2

We can rewrite the premise as

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