Condition for a function to be injective

Thread starter dpa
Start date Oct 2, 2012
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Condition Function Injective

AI Thread Summary

For a function f: X -> Y to be injective, each element in X must map to a unique element in Y, meaning no two different elements in X can share the same image in Y. If there is an element in X that does not have an image in Y, then f cannot be considered a valid function, and thus cannot be injective. The discussion emphasizes that the definition of a function requires every input to have a corresponding output. Therefore, the presence of a preimage without an image disqualifies the function from being injective. Understanding these conditions is crucial for determining the injectivity of a function.

Oct 2, 2012

dpa

Homework Statement

Do all the preimages on X need to have a (and of course I know only one but) image in Y for the f:x->y to be injective?
IS THE FOLLOWING FUNCTION INJECTIVE SINCE ONE ELEMENT OF FIRST DOES NOT HAVE ANY IMAGE

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Homework Equations

The Attempt at a Solution

Thank You.

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Oct 2, 2012

HallsofIvy

Science Advisor

Homework Helper

The way you have shown it, f is not a function from X to Y.

Oct 2, 2012

dpa

Thank you.

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