Thread: Quantifier Project

Hey all, I would really appreciate some help with this problem. I'm not sure what they're asking:

In each of the following cases explain what is meant by the statement and decide whether it is true or false.

(i) For each x in the Integers there exists y in the Integers such that x+y=1

(ii) There exists y in the Integers such that for each x in the Integers, x+y=1

(iii) For each x in the Integers there exists y in the Integers such that xy=x

(iv) There exists y in the Integers such that for each x in the Integers, xy=x


Thanks in advance for the help!

I'm not sure what they're asking

Well, you have to decide if these four statements are true or false. Since the statements are in plain English, not understanding them presents a problem. I am not sure I can help since all I can do is use the same English.

(ii) There exists y in the Integers such that for each x in the Integers, x+y=1

This is false. Indeed, suppose that such y exists. Then x + y = 1 for all integer x, in particular, for x = 0 and x = 1. So, 0 + y = 1, from where y = 1. But then 1 + y = 1 is false since 1 + 1 = 2.

(iii) For each x in the Integers there exists y in the Integers such that xy=x

This is true. Given any x, take y = 1.

Mathematics uses the same words and their meanings as regular language. The phrases "for all" and "there exists" don't cease to mean what they usually mean just because they are used to speak about numbers.