Predicate logic Problem

[Lutz]

Not sure if I'm posting in the right sections, sorry if thats the case

Express in English what each of the following propositions is saying about the set of real numbers, and determine whether they are true or false.

(a)∀x∃y (x X y = x)
(b)∃y∀x (x X y = x)
(c)∀x∃y (x² = y)
(d)∀y∃x (x² = y)

*Capital X means timesed.

I am well and truely stuck on these, if any one can help I would greatly appreaciate it, thank you

 

[Ishuda]

Not sure if I'm posting in the right sections, sorry if thats the case

Express in English what each of the following propositions is saying about the set of real numbers, and determine whether they are true or false.

(a)∀x∃y (x X y = x)
(b)∃y∀x (x X y = x)
(c)∀x∃y (x² = y)
(d)∀y∃x (x² = y)

*Capital X means timesed.

I am well and truely stuck on these, if any one can help I would greatly appreaciate it, thank you

There are certain standard symbols used in mathematics. You probably already know what some of the are, for example put into English the following mathematical statement 2+2=4. Wikipedia has a nice, IMO, list of quite a few of the symbols
http://en.wikipedia.org/wiki/List_of_mathematical_symbols

 

[pka]

(a)∀x∃y (x X y = x)
(b)∃y∀x (x X y = x)
(c)∀x∃y (x² = y)
(d)∀y∃x (x² = y)

You are to translate each of these into the most natural English possible.

a) Every number times some number is unchanged.

b) Some number times any number leaves the second number unchanged.

c) Every number squared is some number.

d) Every number is the square of some number.

Notice how the active and the passive voice functions with the change in variable order.

I will asked you to post which are true or false.

 

[Lutz]

Thank you for the quick replies.


Despite some peoples distrust of wikipedia I often use it as a reference for symbols, wasn't aware of the more extensive list, thank you.


I feel like a idiot now, over complicating the questions, so no wonder I couldn't find any material on it. I assumed it was one continuous sum due to the indentation being only a space.


a) False


b) True (If I'm right in assuming that only the first number is considered to change value and not the second)


c) True


d) False


Thanks, you've really been great help.

 

[pka]

I feel like a idiot now, over complicating the questions, so no wonder I couldn't find any material on it. I assumed it was one continuous sum due to the indentation being only a space.


a) False

a) what about \(\displaystyle y=1~?\)

 

[HallsofIvy]

The difference between (a) and (b), "for all numbers, x, there exist a number y, such that xy= x" and "there exist a number, y, such that, for all numbers, x, xy= x" is that (a) leaves open the possibility that the "y" is different for every x while (b) asserts that the same y applies to all x. But taking y= 1 satisfies both since the possibility that y is different for every x does not mean it must be.