Mohammed Mujamil
New member
- Joined
- Jun 28, 2021
- Messages
- 1
"As I understand it, the '⇒' symbol (which stands for the word 'implies') means the following: Given two statements A and B:
( ( if (A == true) then (B == true) )
or
(if (A == false) then ((B == true) or (B == false))) )
or in words: If Statement A is true, B must also be true, though if A is false, B can be either true or false.
The definition of an injective function according to my book and wikipedia is: f is called injective, if the following holds true: f(a)=f(b)⇒a=b Which means if we apply two elements a,b∈A to f:A→B, and the resulting two elements f(a) and f(b)∈B are equal, then the original two values a,b must be the same element, hence a=b.
Now comes the part that confuses me: If we apply the definition above to the other, we get the following:
if f(a)=f(b) then a=b
or
if f(a)≠f(b) then a=b or a≠b
So the last line says if f(a)≠f(b) then a and b can either be the same element or different elements. The latter is fine, if f returns two unique elements, the elements a,b applied to the function must also be unique, hence a≠b. But its also possible for a=b, which would mean that it is possible to get more than one unique element from f, according to the definition.
That last conclusion is of course wrong, since that would be a property of a surjective function and not an injective one. Thanks for your time."
( ( if (A == true) then (B == true) )
or
(if (A == false) then ((B == true) or (B == false))) )
or in words: If Statement A is true, B must also be true, though if A is false, B can be either true or false.
The definition of an injective function according to my book and wikipedia is: f is called injective, if the following holds true: f(a)=f(b)⇒a=b Which means if we apply two elements a,b∈A to f:A→B, and the resulting two elements f(a) and f(b)∈B are equal, then the original two values a,b must be the same element, hence a=b.
Now comes the part that confuses me: If we apply the definition above to the other, we get the following:
if f(a)=f(b) then a=b
or
if f(a)≠f(b) then a=b or a≠b
So the last line says if f(a)≠f(b) then a and b can either be the same element or different elements. The latter is fine, if f returns two unique elements, the elements a,b applied to the function must also be unique, hence a≠b. But its also possible for a=b, which would mean that it is possible to get more than one unique element from f, according to the definition.
That last conclusion is of course wrong, since that would be a property of a surjective function and not an injective one. Thanks for your time."