discrete math: The logic of quantified statements : ambiguity concerning for all...

[studentMCCS]

Something is bothering me about the section in my discrete math book about the logic of quantified statements. I asked my teacher today, and he couldn't seam to figure it out either.

We have been negating quantified statements, showing the converse, contrapositive, and inverse.

We have been considering:

If n is any prime number that is greater than 2, then n+1 is even,

to be equivalent to:

for all prime numbers p, if p is greater than 2, then p+1 is even.

However, when it comes to negating, getting the converse etc there seam to be ambiguous differences.

For example, the converse of

statement: If n is any prime number that is greater than 2, then n+1 is even.

Converse: if n+1 is even, then n is a prime number greater than two.

Whereas, the converse of the for all, would be,

statement: for all prime numbers p, if p is greater than 2, than p+1 is even.

converse: for all prime numbers p, if p+1 is even, than p is greater than 2.

So basically, the two forms are not equivalent?

 

[pka]

In writing formally, in the class, we have been considering:
If n is any prime number that is greater than 2, then n+1 is odd,

to be equivalent to:

for all prime numbers p, if p is greater than 2, then p is odd.

You have misstated this problem.
Those two statements are not equivalent.

The first of the two is false.
This is true: If n is a prime number greater than two then n+1 is even.

 

[studentMCCS]

You have misstated this problem.
Those two statements are not equivalent.

The first of the two is false.
This is true: If n is a prime number greater than two then n+1 is even.

Your right, sorry about that, I've made the corrections.

statement 1: for all prime numbers p, if p is greater than 2, then p+1 is even.

statement 2: in n is any prime number greater than 2, then n+1 is even.

 

[pka]

For example, the converse of
statement: If n is any prime number that is greater than 2, then n+1 is even.
Converse: if n+1 is even, then n is a prime number greater than two.

Whereas, the converse of the for all, would be,

statement: for all prime numbers p, if p is greater than 2, than p+1 is even.

converse: for all prime numbers p, if p+1 is even, than p is greater than 2.

The converse of "If n is any prime number that is greater than 2, then n+1 is even." is "For any prime n, if n+1 is even then n is greater than two.

 

[studentMCCS]

The converse of "If n is any prime number that is greater than 2, then n+1 is even." is "For any prime n, if n+1 is even then n is greater than two.

That is what I had though, but my book says otherwise. The question is to show that the converse of the true statement, If n is any prime number that is greater than 2, then n+1 is even, is false. In the back of the book the answer is that the converse is, if n+1 is even, then n is a prime number greater than two.

 

[pka]

That is what I had though, but my book says otherwise. The question is to show that the converse of the true statement, If n is any prime number that is greater than 2, then n+1 is even, is false. In the back of the book the answer is that the converse is, if n+1 is even, then n is a prime number greater than two.

The converse of "If n is prime and greater than two then n+1 is odd." is "If n+1 is even then n is prime and greater than two."
I agree with that. However not with the way the original statement is quantified. I think it is a poorly composed question. What text is it? Chances are good that I have it.

 

[studentMCCS]

The converse of "If n is prime and greater than two then n+1 is odd." is "If n+1 is even then n is prime and greater than two."
I agree with that. However not with the way the original statement is quantified. I think it is a poorly composed question. What text is it? Chances are good that I have it.

The text is "Discrete Mathematics with Applications, Fourth Edition by Susanna S. Epp. The problem is section 3.2, 25 a.

 

[pka]

The text is "Discrete Mathematics with Applications, Fourth Edition by Susanna S. Epp. The problem is section 3.2, 25 a.

I don't have that one. I had it at one time.
I am surprised. One normally does not treat a quantifier that way.

 

[studentMCCS]

I don't have that one. I had it at one time.
I am surprised. One normally does not treat a quantifier that way.

The question had my professor pretty thrown off this morning. Thanks for your help!