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Functions problem

ranga519

New member
May 19, 2018
5
Let f be a function from a set X to a set Y, moreover, A ⊆ X. What comparison sign canput instead? to assert "f ^(−1) (f(A)) ? A" become true? (Possible signs of comparison in this : ⊆, ⊇, =. It is necessary to take into account all options.

f ^(−1) - inverse of fall options.), Let f be a function from a set X to a set Y, moreover, A ⊆ X. What comparison sign canput instead? to asserte-one(f (a))?become true? (Possible signs of comparison in this and the following two problems: ⊆, ⊇, =. It is necessary to take into accountall options.)
 

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
368
Let f be a function from a set X to a set Y, moreover, A ⊆ X. What comparison sign canput instead? to assert "f ^(−1) (f(A)) ? A" become true? (Possible signs of comparison in this : ⊆, ⊇, =. It is necessary to take into account all options.


Suppose $x\in A$. Then $f(x)\in f(A)$ so that $x\in f^{-1}(f(A))$. But there might exist $y\notin A$ such that $f(y)\in f(A)$. That y would also be in $f^{-1}(f(A))$. So what we can say is that $A\subseteq f^{-1}(f(A))$ with the "=" possible but not necessarily. For example if f is a "constant function", $f(x)= y\in Y$ for all $x\in X$ where y is a specific member of Y, then $f^{-1}(f(A))= X$ for A any subset of X.