Reviewing for exam, confused on this universal/exists true/false

[mr_coffee]

Hello everyone.

Exam time is coming and I'm review, he let us see the answer key to a sample exam but I'm still not sure how they came up with the following:

If the variables a and b range over the set of postive integers, then is each of the following statements true or false, circle ur answers.

http://suprfile.com/src/1/3p2it9s/lastscan.jpg



1. For all a there exsits a b such that a < b. yes i see this is true,
2. For all b there exists an a such that a < b. But I'm confused on why this isn't true...or is it false based on the answer above, if for all a there exists b such that a < b?

3. There exists an a for all b such that a < b, I think i see why this is false, its a strong statement to say. if a = 5 b = 5, then this is false

4. There exists an a for all b such that a <= b, this is true becuase if u let a = b, then its always true.

5. There exists a b for all a such that a <= b, i don't see why this is also not true though.

Can someone explain to me why they are true and why they are false exactly?


Thanks!

 

[quasar987]

The first thing to do is to try to find a counter exemple.

2. It's true for all b except b=1. Because there are no positive integer smaller than 1.

3. The statement reads, "There exists and a such that for all b, a<b". You misplaced the words "such that", which completely changes the significance of the statement. You made this "error" in interpreting 4. and 5. as well, so no wonder you're confused about the answers.

Try again, with the right interpretation and tell me if you still have problems.

 

[AKG]

I will use "PI" to mean "positive integer".

1 says "for each PI, there is a PI bigger than it"
2 says "for each PI, there is a PI less than it"
3 says "there is a PI less than any PI"
4 says "there is a PI less than or equal to any PI"
5 says "there is a PI greater than or equal to any PI"

Can you see that the above are the correct interpretations? Given these interpretation, can you see why only 1 and 4 are true?

 

[mr_coffee]

Thanks for the responces guys but I'm still confused on the #4.

There is a postive integer less than or equal to any postive integer...
well, if its less than or EQUAL too, even if b = 1, you could let a = 1, and this would make that statement true wouldn't it?

And for the last one, since b comes first is that why you said,
there is a PI greater than or equal to any PI, because u look from right to left rather than left to right with a <= b?

Thanks again!

 

[quasar987]

Thanks for the responces guys but I'm still confused on the #4.

There is a postive integer less than or equal to any postive integer...
well, if its less than or EQUAL too, even if b = 1, you could let a = 1, and this would make that statement true wouldn't it?

One of us has severly impared eyesight, because I see that TRUE is circled for #4!

 

[mr_coffee]

Oops I'm not thinking very well today...
I can see fine but the amount of starbucks I've consumed is affecting me i think.

That brings up another question, i do see why those 2 are true, but why is #5 false?

5 says "there is a PI greater than or equal to any PI"

Well can't u just let b = a? then it would be always true becuse for any postive integer, you can set b or a equal to that number like you can for less than or equal to, #4.

 

[quasar987]

#5 says "There exists a PI that is greater to any PI". It means "You can find one PI that is greater than any PI."

"Setting" b=a means choosing a different 'b' for every 'a'. But the statement says that there is ONE 'b' that satisfies $b\geq a$ for ALL a.

 

[mr_coffee]

Oooo i think i see now, becuase postive integers are endless, you can keep going and going and never find an end, but becuase integers can only go from 1 to whatever, that's why the less than or equal to statement was true, while this one is false because u can always find a number that is b+1 THank you!