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f:U->V; f is injective, for a,b in U

1. a!=b implies f(a)!=f(b)

2. f(a)=f(b) implies a=b

Why isn't the definition:

3. a!=b if and only if f(a)!=f(b)

similarly,

4. a=b if and only if f(a)=f(b)

From 1, if a!=b implies f(a)!=f(b); consider a=b; certainly f(a)=f(b); so why isn't it that a!=b if and only if f(a)!=f(b).

What is the logic behind the definition of injectivity?