If and only if versus if .... then

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In summary, Herb Gross uses the expressions "if and only if" and "if ... then" in his supplementary notes to discuss different implications between propositions. While they may seem interchangeable, they actually represent different directions of implication and serve different purposes. Understanding their usage is important for comprehending more complex arguments in his lectures.
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inthenickoftime
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Author seems to use both in similar/identical context.
I found a really nice set of lectures by Herb Gross on the subject of calculus, but I'm confused by his usage of if and only if and if ... then in his supplementary notes.

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It seems to me that he describes identical situations on both pages, but in one case the author uses the wording "if and only if" while on the other page, an "if ... then" expression is found. Is he using them interchangeably here? Why did he change the wording the second time around? I need to make sure I understand his usage of underlined (red) expressions otherwise I won't be able to follow more elaborate arguments that will follow.
 
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inthenickoftime said:
It seems to me that he describes identical situations on both pages, but in one case the author uses the wording "if and only if" while on the other page, an "if ... then" expression is found.
Yes, the two situations seem identical to me
inthenickoftime said:
Is he using them interchangeably here?
I don't think so, even if the situations seem identical. Rather, I think he's just focusing on one direction, if .. then.

Probably you know, but for statements/propositions p and q, ##\text p \Leftrightarrow \text q## means the following two implications:
##\text p \Rightarrow \text q## and ##\text q \Rightarrow \text p##. On the first page, he's talking about the iff statement. On the second page he's talking about one of the latter pair of implications to make a point. I don't believe his intent is that they are interchangeable.
 
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inthenickoftime said:
I found a really nice set of lectures by Herb Gross on the subject of calculus, but I'm confused by his usage of if and only if and if ... then in his supplementary notes.
Note that: "if ##A## then ##B##"means "##A \Rightarrow B##" or "##A## only if ##B##".

And "##B \Rightarrow A##"means "##A## if ##B##".
 
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Related to If and only if versus if .... then

1. What is the difference between "if and only if" and "if...then" statements?

"If and only if" statements are used to show that two statements are logically equivalent, meaning that they have the same truth value. In other words, if one statement is true, then the other must also be true. On the other hand, "if...then" statements show a conditional relationship, where the truth of the first statement guarantees the truth of the second statement, but the truth of the second statement does not necessarily guarantee the truth of the first.

2. When should "if and only if" be used instead of "if...then"?

"If and only if" should be used when you want to express that two statements are equivalent and both must be true in order for the overall statement to be true. This is often used in mathematical proofs or logical arguments.

3. Can "if and only if" be replaced with "if...then" in all cases?

No, "if and only if" cannot always be replaced with "if...then". "If and only if" statements are bi-directional, meaning that the truth of one statement guarantees the truth of the other, and vice versa. "If...then" statements are one-directional, meaning that the truth of the first statement guarantees the truth of the second, but not the other way around.

4. How is "if and only if" represented in symbolic logic?

In symbolic logic, "if and only if" is represented by the double arrow symbol (↔). This symbol is placed between the two statements that are being shown to be equivalent.

5. Are there any real-life examples of "if and only if" statements?

Yes, there are many real-life examples of "if and only if" statements. For instance, "you will pass the test if and only if you study hard" means that studying hard is both necessary and sufficient for passing the test. Another example could be "you can enter the club if and only if you are over 21 years old", where being over 21 is both necessary and sufficient for entering the club.

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