- #1
CAF123
Gold Member
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I seem to have a couple of contradictory statements of what a countable set is defined to be:
In my textbook I have:
'Let E be a set. E is said to be countable if and only if there exists a 1-1 function which takes ##\mathbb{N}## onto E.'
This implies to me that that there has to exist a beijection from ##\mathbb{N}## to E since they use 1-1 and onto in the defintion.
In my notes I have: E is said to be countable if there is a bijection ##f: \mathbb{N} \mapsto E##'
However, when I looked on Wikipedia, I read:
A set S is called countable if there exists an injective function f from S to the natural numbers N.
Is there a contradiction here and if so, what is the true defintion?
In my textbook I have:
'Let E be a set. E is said to be countable if and only if there exists a 1-1 function which takes ##\mathbb{N}## onto E.'
This implies to me that that there has to exist a beijection from ##\mathbb{N}## to E since they use 1-1 and onto in the defintion.
In my notes I have: E is said to be countable if there is a bijection ##f: \mathbb{N} \mapsto E##'
However, when I looked on Wikipedia, I read:
A set S is called countable if there exists an injective function f from S to the natural numbers N.
Is there a contradiction here and if so, what is the true defintion?