MHB Identity Function: Definition, Examples & Properties

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The identity function is defined as a function that maps every element of a set to itself, denoted as $id_A: A \to A$. It is characterized as a bijection, meaning it is both injective and surjective. The composition of the identity function with any other function $f$ results in the original function, expressed as $f \circ id_A = f$. The discussion highlights that the identity function maintains the properties of bijections when composed with other functions. Understanding the identity function is crucial for grasping more complex function properties in mathematics.
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dana said:
hey,
question is attached

thanks in advance!

Hi dana!

Well... since $id_A: A\to A$ is a bijection, it seems fair to me that $f \circ g: A \to A$ given by $f \circ g: a \mapsto a$ is also a bijection...
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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