Understanding Nested Quantifiers: How to Determine Truth Value | Help & Tips

[Bucs44]

Here is the problem that I'm having trouble solving - I'm not sure where to begin. I need to determine the truth value but don't know how to do that.

Ax3y(x^2 < y + 1)

 

[matt grime]

Perhaps you could write that out in plain english. It might even help you understand how to solve your problem.

 

[Bucs44]

For every real number x there exists y, x to the second power is less than y plus 1.

So basically I need to find a number that is less than y + 1

 

[matt grime]

No. Read it out loud inserting all of the words.

For all x there is a y such that the condition x^2<y+1 is true.

 

[HallsofIvy]

In other words, to prove that true, you must prove that for any given x there exist y such that y+ 1>x². You need to show that, whatever x is you can find a corresponding y.

 

[Bucs44]

So if I stated x = 1, then x2 would be 1 and then would or could I say y is 1 as well? Making y + 1=2

 

[matt grime]

Now how can you do this for any x?

 

[Bucs44]

That's what I don't get - I'm not sure what numbers I'm supposed to be inputing here. Please help as I'm really confused

 

[Bucs44]

After working on this - Is this the answer?

y = x2 + 2x = (x + 1)2 – 1
x < 0 then (x – 1)2 > x2 so y can be x2 – 2x -2
x = 0 then y can be 0

 

[matt grime]

You're not supposed to put any numbers in that's the point.

Let's play a game. I'm thinking of a number x. Can you give me a number y(possibly in terms of x) so that y+1 is definitely larger than x^2?

 

[Bucs44]

How about z?