Proving Mathematical Statements: a Real-Life Example

In summary, the two statements provided mean the same thing and can be proven false by setting x = 2. However, the order of the quantifiers in the statements makes a difference in the truth value. The first statement is true because for every real x, there exists a real y that satisfies the equation. The second statement is false because there exists a real y that does not satisfy the equation for every real x.
  • #1
Rob Hal
13
0
Hi all,

If I have these two statements given to me, and I have to determine whether they are true or not.

a) [tex] \forall x \epsilon R [/tex] [tex]\exists y \epsilon R [/tex] [tex](y^2 = x^2 + 1)[/tex]
b) [tex]\exists y \epsilon R [/tex] [tex]\forall x \epsilon R [/tex] [tex](y^2 = x^2 + 1) [/tex]

Now, to me, they both mean exactly the same thing, and both can be shown to be false by setting x = 2, then y is not a real number.

However, seeing that the question specifically asks to prove just those two statements, I'm wondering if perhaps I am interpreting them wrong and they actually mean two different things.

Thanks in advance for any advice,
Robbie
 
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  • #2
Rob Hal said:
Now, to me, they both mean exactly the same thing, and both can be shown to be false by setting x = 2, then y is not a real number.
If x = 2 then [itex]y^2 = 2^2 + 1 \Leftrightarrow y = \pm \sqrt 5 [/itex]. Those are real numbers, no?
 
  • #3
lol... yeah...
I was thinking I was looking for rationals only... whoops...

Still, is there any difference in the two statements themselves?
 
  • #4
Try a simpler one to see how the order of the quantifiers makes a difference:

[tex]\forall x \in \mathbb{R} \ \exists y \in \mathbb{R} \ (x = y)[/tex]

This says that for every real x that I choose, I can find at least one real y that is equal to that x. This is obviously true, since x = x.

[tex]\exists y \in \mathbb{R} \ \forall x \in \mathbb{R} \ (x = y)[/tex]

This says that I can find at least one real y that is equal to every real x. Well, there's more than one real number, so this is false.
 
Last edited:

What is the importance of proving mathematical statements in real-life examples?

Proving mathematical statements in real-life examples is important because it allows us to validate the accuracy and applicability of mathematical concepts in practical situations. This helps to build trust and confidence in the validity of mathematical principles.

What is the process of proving a mathematical statement in a real-life example?

The process of proving a mathematical statement in a real-life example involves first identifying the statement to be proven, understanding the context of the real-life situation, and then using logical reasoning and mathematical techniques to demonstrate the truth of the statement.

What are some common challenges when proving mathematical statements in real-life examples?

Some common challenges when proving mathematical statements in real-life examples include dealing with complex and abstract concepts, finding appropriate real-life examples that accurately reflect the statement, and ensuring the validity and reliability of the data and measurements used.

How does proving mathematical statements in real-life examples contribute to scientific research?

Proving mathematical statements in real-life examples is an essential part of scientific research as it provides a rigorous and verifiable way to test and validate mathematical theories and models. This helps to support the development of new scientific knowledge and advancements.

Are there any limitations to proving mathematical statements in real-life examples?

While proving mathematical statements in real-life examples is a valuable tool, it is important to note that not all real-life situations can be accurately represented by mathematical models. Additionally, the complexity and variability of real-life scenarios can make it challenging to draw definitive conclusions from mathematical proofs alone.

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