0

I'm struggling to understand the use of "if" in different context in mathematics.

Firstly, when we use "if" in a statement, like "if a, then b". It implies that if b, then not necessarily a.

However, in the notes of a professor, he wrote that the definition of an injective function as such:

Let F : A $\rightarrow$ B,

F is injective if F($a_1$) $=$ F($a_2$) only if $a_1 = a_2$

In a proof later on, he uses if F($a_1$) $=$ F($a_2$) when $a_1 \neq a_2$ to show F isn't injective. Does this mean that in the context of a definition, "if" is the same as "if and only if"?

Heng Wei
  • 419
  • 2
    It's common in definitions to use "if" where one would normally use "iff." (Is it a good idea to do that? I'm ambivalent about it, personally.) – anomaly Jun 15 '22 at 17:01
  • 4
    Yes. It is a community convention that definitions are always if-and-only-if statements—indeed what use would they be if they weren't? ("A number is even if it is a multiple of 2" would be useless if maybe some other numbers were even too.) In definitions it's common to write "if" as a shorthand, but definitions are always if-and-only-if. – Greg Martin Jun 15 '22 at 17:01

0 Answers0