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

  • Thread starter Bucs44
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In summary, the problem presented is about determining the truth value of the statement "For every real number x there exists y, x to the second power is less than y plus 1." The conversation discusses how to interpret and solve this statement, with the overall goal being to find a way to prove that the statement is true for any given x. The conversation also includes an example using specific values for x and y to illustrate the concept.
  • #1
Bucs44
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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)
 
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  • #2
Perhaps you could write that out in plain english. It might even help you understand how to solve your problem.
 
  • #3
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
 
  • #4
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.
 
  • #5
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.
 
  • #6
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
 
  • #7
Now how can you do this for any x?
 
  • #8
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
 
  • #9
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
 
  • #10
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?
 
  • #11
How about z?
 

1. What is a nested quantifier?

A nested quantifier is a logical expression that contains multiple quantifiers, such as "for all" and "there exists", within its structure. This allows for the creation of complex statements that involve multiple variables and predicates.

2. How do I determine the truth value of a statement with nested quantifiers?

To determine the truth value of a statement with nested quantifiers, you must first evaluate the innermost quantifier and then work your way outwards. Use the rules of quantifiers and logical equivalences to simplify the statement and determine its truth value.

3. What is the difference between nested quantifiers and multiple quantifiers?

Nested quantifiers are quantifiers that are contained within another quantifier, while multiple quantifiers are quantifiers that are used together to express a statement. Nested quantifiers can lead to more complex statements and require a different approach to determine their truth value.

4. Can I interchange the order of nested quantifiers?

No, the order of nested quantifiers cannot be interchanged. The placement of quantifiers is crucial in determining the meaning and truth value of a statement. Changing the order of nested quantifiers can lead to a different statement with a different truth value.

5. How can I use nested quantifiers to express mathematical statements?

Nested quantifiers can be used to express mathematical statements involving multiple variables and predicates. For example, the statement "For every real number x, there exists a real number y such that x + y = 10" can be written using nested quantifiers as "∀x∃y (x + y = 10)". This allows for a more concise and precise representation of mathematical statements.

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