Definition of injectivity

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Instead, the definition simply states that if a and b are different, then their outputs must also be different (a!=b implies f(a)!=f(b)). This definition is sufficient to ensure that each input has a unique output, and therefore the function is injective. In summary, the definition of injectivity states that for a function f: U->V, f is injective if and only if for any inputs a and b in U, if a is not equal to b, then f(a) is not equal to f(b). This definition ensures that each input has a unique output.
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I have a concern about the definition of injectivity:

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?
 
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  • #2
studious said:
I have a concern about the definition of injectivity:

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?
Since "if a= b then f(a)= f(b)" is part of the definition of "function", it not necessary to include it in the definition of "injective function".
 
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1. What is the definition of injectivity?

Injectivity is a property of a mathematical function in which each element in the range of the function is mapped to by only one element in the domain. In simpler terms, it means that different inputs will always produce different outputs.

2. How is injectivity different from surjectivity and bijectivity?

Injectivity is just one aspect of a function's behavior, while surjectivity and bijectivity are two other properties. Surjectivity means that every element in the range is mapped to by at least one element in the domain, while bijectivity means that a function is both injective and surjective.

3. Can a function be injective if it has multiple outputs for the same input?

No, a function must have unique outputs for each input in order to be injective. If multiple outputs exist for the same input, then the function is not one-to-one and therefore not injective.

4. How can I determine if a function is injective?

To determine if a function is injective, you can use a vertical line test. If a vertical line drawn on the graph of the function intersects the graph at more than one point, then the function is not injective.

5. What is the importance of injectivity in mathematics?

Injectivity is important because it allows us to determine whether a function has an inverse. If a function is injective, it has a unique inverse, meaning that we can "undo" the function and retrieve its original input from the output. This is useful in many applications, such as cryptography and data compression.

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