Set operations and mappings of sets by functions

Given a set $X$ , for each $x\in X$ , we use $t(x), t_1(x), t_2(x)$ etc., to denote truth functions on $X$ . That is, for each value of $x\in X$ , $t(x), t_1(x), t_2(x)$ etc., take values true or false. For example, $x < 3$ and $|f(x)| < 1$ for some real-valued function function $f$ are examples of truth functions.

Now, we have the following identities which are helpful in reducing intermediate steps of many problems in analysis. (Here, we just write $t(x)$ to denote that $t(x) = true$ )

Union
$\{x\in X\mid t_1(x)\mbox{ or }t_2(x)\} = \{x\in X|t_1(x)\}\bigcup\{x\in X|t_2(x)\}$

Intersection
$\{x\in X\mid t_1(x)\mbox{ and }t_2(x)\} = \{x\in X\mid t_1(x)\}\bigcap\{x\in X\mid t_2(x)\}$

Negation
$\{x\in X\mid t(x)\mbox{ is not true }\} = X \thicksim \{x\in X\mid t(x)\} = \mbox{ Compliment of the set }\{x\in X\mid t(x)\}\mbox{ in }X$

More generally, we have the following formulae.

Countable union
$\{x\in X \mid \exists n\in\mathbb{N}, t_n(x)\}$ can be written as $\{x\in X \mid t_1(x)\mbox{ or }t_2(x)\mbox{ or }t_3(x)\mbox{ or }\cdots \}$

Hence, $\{x\in X\mid \exists n\in\mathbb{N}, t_n(x)\} = \bigcup \limits_{n\in\mathbb{N}}\{x\in X\mid t_n(x)\}$ .

Countable intersection
$\{x\in X\mid \forall n\in\mathbb{N}, t_n(x)\}$ can be written as $\{x\in X\mid t_1(x)\mbox{ and }t_2(x)\mbox{ and }t_3(x)\mbox{ and }\cdots \}$

Hence, $\{x\in X\mid \forall n\in\mathbb{N}, t_n(x)\} = \bigcap \limits_{n\in\mathbb{N}}\{x\in X\mid t_n(x)\}$ .

Thus, for example, given a sequence of functions $\{f_n:X\to\mathbb{R}\}$ ,

(Some typos in the following need to be fixed)

$\begin{aligned} \{x\in X \mid |f_j(x) - f_k(x)| N\} & = \{x\in X \mid \forall j > N, |f_j(x) - f_k(x)| N\} \\ &= \bigcap \limits_{j=N}^\infty\{x\in X \mid \forall k > N, |f_j(x) - f_k(x)| < \epsilon\}\\ &= \bigcap \limits_{j=N}^\infty\bigcap \limits_{k=N}^\infty\{x\in X \mid |f_j(x) - f_k(x)| < \epsilon\} \end{aligned}$

Mappings of sets from range to domain
Given a function $g:X\to Y$ , we have the following identities:
* For all $A,B\subseteq Y, g^{-1}(A\cup B) = g^{-1}(A) \cup g^{-1}(B)$
* For all $A,B\subseteq Y, g^{-1}(A\cap B) = g^{-1}(A) \cap g^{-1}(B)$
* For all $A\subseteq Y, g^{-1}(Y \thicksim A) = X \thicksim g^{-1}(A)$

These formulae follow from the identities given earlier to handle unions, intersection and negations:

$\begin{aligned} g^{-1}(A\cap B) & = \{x\in X\mid g(x)\in A\cap B\} \\ & = \{x\in X\mid g(x)\in A\mbox{ and }g(x)\in B\} \\ & = \{x\in X\mid g(x)\in A\}\cap \{x\in X|g(x)\in B\} \\ & = g^{-1}(A)\cap g^{-1}(B) \end{aligned}$

Caution

While applying the formulae Union through Countable Intersection above, it is important for the left hand side to be in the form given. That is, the truth functions must be functions of elements of target set. For example, in case of Union, the left hand side is $\{x\in X\mid t_1(x)\mbox{ or }t_2(x)\}$ . Here, the target sets have elements $x\in X$ and the truth functions $t_1$ and $t_2$ are functions of $x$ . Thus, given any value of $x\in X$ , $t_1(x)$ and $t_2(x)$ must result in one unique value in $\{true, false\}$ .

An example illustrates a possible incorrect use of these formulae. In general, $g(A\cap B)\neq g(A)\cap g(B)$ . However, since $g(A\cap B) = \{y=g(x)\in Y|x\in A\mbox{ and }x\in B\}$ , one may incorrectly try to apply Intersection formula here. Observe that the target set has elements $y\in Y$ , while the truth functions are functions of $x$ . To apply the formulae, the truth functions must be expressed as functions of $y$ .

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