Question about if and then statements. IE implication statements.

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In summary, "If and then" statements, also known as implication statements, are logical statements in the form of A \Rightarrow B, where A is the hypothesis and B is the conclusion. These statements are used to prove a result by assuming the truth of the hypothesis and showing that the conclusion must also be true. This can be done by using previous statements or assumptions to arrive at the conclusion. It is also important to note that different symbols, such as →, may be used to represent this type of statement. Additionally, studying logic can be helpful in understanding and solving mathematical problems.
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kramer733
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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?
 
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  • #2


kramer733 said:
I say yes that the "if" is the hypothesis. Then the "then" is what we are trying to prove right?

Right!
 
  • #3


micromass said:
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?
 
  • #4


kramer733 said:
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.
 
  • #5


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,

[itex]A \Rightarrow B[/itex]

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.
 

1. What is an implication statement?

An implication statement, also known as an "if and then" statement, is a logical statement that links two phrases using the words "if" and "then". The first phrase, known as the antecedent, is the condition that must be met for the second phrase, known as the consequent, to be true.

2. How is an implication statement represented in symbolic logic?

In symbolic logic, an implication statement is represented by the symbol "→". For example, the statement "If it rains, then the ground is wet" can be represented as "P → Q", where P represents "it rains" and Q represents "the ground is wet".

3. What is the difference between an implication statement and a biconditional statement?

An implication statement only requires the antecedent to be true in order for the consequent to be true, while a biconditional statement requires both the antecedent and the consequent to be true for the statement to be true. In other words, an implication statement is a one-way relationship, while a biconditional statement is a two-way relationship.

4. Can an implication statement be false if the antecedent is true?

No, if the antecedent of an implication statement is true, then the statement must be true. This is because the consequent is only relevant if the antecedent is true. If the antecedent is false, then the truth value of the consequent is irrelevant.

5. How is an implication statement used in scientific research?

In scientific research, implication statements are commonly used to state hypotheses and make predictions. For example, a researcher might hypothesize that "If a person exercises regularly, then their risk of heart disease will decrease". This implication statement can then be tested through experiments and data analysis.

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