- #1
lonewolf5999
- 35
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I'm trying to learn some analysis on my own, and as this is the first proof-based book I'm reading, I have a basic question about definitions I was hoping someone could help me with. For example, the book I'm reading says that: Given a subset of the real numbers A, b is an upper bound of A if every element a of A is less than or equal to b. I rephrase this definition of the upper bound as: if every element a of A is less than or equal to b, then b is an upper bound of A.
My question is: is this definition an if and only if statement? That is, is the statement "If b is an upper bound of A, then every element a of A is less than or equal to b" also true? It seems like it should be, but I was hoping to get some confirmation or clarification on this. More generally, is it safe to assume that all definitions are if and only if statements? If not, is there any way to tell when they aren't?
My question is: is this definition an if and only if statement? That is, is the statement "If b is an upper bound of A, then every element a of A is less than or equal to b" also true? It seems like it should be, but I was hoping to get some confirmation or clarification on this. More generally, is it safe to assume that all definitions are if and only if statements? If not, is there any way to tell when they aren't?