Math logic: predicates and quantifiers

[Handteg]

I just want to know if I reached the correct conclusion. My conclusion was that the statement was true because r could just equal the reciprocal. Or did I make a mistake in understanding the statement? Does the statement mean that there is a single r, that when multiplied by any x, the statement is true? The problem is 8a. I have attached a photo of the exercise

 

[Dr.Peterson]

I just want to know if I reached the correct conclusion. My conclusion was that the statement was true because r could just equal the reciprocal. Or did I make a mistake in understanding the statement? Does the statement mean that there is a single r, that when multiplied by any x, the statement is true? The problem is 8a. I have attached a photo of the exercise

Let's make it readable:


Try writing out what the whole statement means in English, before answering the question:

For all x, ...​

This doesn't start with "a single r"; that comes later. Finish this, and we can discuss it.

 

[pka]

Ask your self if the set of natural numbers is bounded above? Is the set of real numbers?
Does every real number have a multiplicative inverse?

 

[Handteg]

Let's make it readable:
View attachment 21944

Try writing out what the whole statement means in English, before answering the question:

For all x, ...​

This doesn't start with "a single r"; that comes later. Finish this, and we can discuss it.

For all x there exists at least on one r such that x•r=1 is true.

 

[pka]

For all x there exists at least on r such that x•r=1 is true.

What if \(\large x=0~?\)

 

[Handteg]

Yes, you’re absolutely right haha. I can’t believe I didn’t consider 0.