Question about if and then statements. IE implication statements.

[kramer733]

Question about "if and then" statements. IE implication statements.

Homework Statement

When something is for example asking for:

if |x-3|<δ, prove that |x+3| <δ + k (where k is a constant)

are they supposing it's true? Like are they giving you a hypothesis? How do implication statements work?

Homework Equations

The Attempt at a Solution

I say yes that the "if" is the hypothesis. Then the "then" is what we are trying to prove right?

 

[micromass]

I say yes that the "if" is the hypothesis. Then the "then" is what we are trying to prove right?

Right!

 

[kramer733]

Right!

Thanks for the verification. Now here's another question that I'm confused about.

The question states the following:

Let f(x) = 1/(x?2) and c = 3.
If x>2, prove that |f(x) - f(3)| <|x^2 - 9|/36.
if |x-3|<δ, prove that |x+3| <δ + k (where k is a constant, which you must determine).
if δ ≤ 1 and |x - 3| < δ, use these results to find M, such that |f(x) - f(3)| < M*δ


Now for the 3rd "if" statement, that means I'm allowed to use the previous statements I've proved to prove this one right? So I'm not working in a "vacuum". Is that correct?

 

[micromass]

Thanks for the verification. Now here's another question that I'm confused about.

The question states the following:

Let f(x) = 1/(x?2) and c = 3.
If x>2, prove that |f(x) - f(3)| <|x^2 - 9|/36.
if |x-3|<δ, prove that |x+3| <δ + k (where k is a constant, which you must determine).
if δ ≤ 1 and |x - 3| < δ, use these results to find M, such that |f(x) - f(3)| < M*δ


Now for the 3rd "if" statement, that means I'm allowed to use the previous statements I've proved to prove this one right? So I'm not working in a "vacuum". Is that correct?

Correct! Once you proved the two other if-statements, then you are allowed to use them.

 

[5hassay]

Even though it seems like this is already solved...

If you have a statement, say, A, and another statement, say, B, and if the logic is such that, "If A, then B," we assume A is true. Of course, if you are given a problem that requires you to prove that this logical statement is true, you can use your assumption of the truth of A to prove that B is a consequence of A. In general, an "If... then..." statement is symbolized,

A \Rightarrow B

for some statements A, B. Again, this means that if A is true, then B is so/is true/is a result/is a consequence of A. Also, different symbols are wildly used, I think, for this same statement, such as →. And, on an extended note, I would suggest looking up "logic" on the internet or something, as its interesting! For example, you will likely encounter many more logical statements in mathematics, and find that taking the 'opposite' of statements you want to show can help to actually prove them.