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So what we have so far is that any and all subsets are implied by a set. If there exist a set, then all the subsets within it are implied to exist also. This includes the elements of a set. The elements of a set are implied by the existence of a set.

One of the most natural things to do with sets is to count the number of elements it has. This can be done in order to give a relative size to subsets. So the number of elements works as a measure of a set.

You can count an element as a member of a set only if the element is included in the set. If we label the elements with arbitrary coordinates, we can describe this mathematically as,

[tex]\[\int_{\rm{A}} {{\rm{\delta (x - x}}_0 ){\rm{dx}}} \, = \,1\][/tex]

The integral is 1 if [tex]\[{\rm{x}}_{\rm{o}} \in \,\,{\rm{A}}\][/tex]. In other words, if [tex]\[{\rm{A}} \to {\rm{x}}_0 \][/tex] is true, then the integral is 1. But if [tex]\[{\rm{A}} \to {\rm{x}}_0 \][/tex] is not true, then the integral is 0. The integral is 1 or 0 as the implication is true or false. The integrand is the dirac delta "function".

Now it seems to me that the above is all true no matter what the size of the region A is. We can let A become smaller and smaller and shrink down to the size of another point, x, and the above is still true. This procedure would be the same as taking the derivative of an integral, which is the same as the integrand, [tex]\[{{\rm{\delta (x - x}}_0 )}\][/tex]. If [tex]\[x \to {\rm{x}}_0 \][/tex] is true then [tex]\[{\rm{\delta (x - x}}_0 ) \ne 0\][/tex]. And if [tex]\[x \to {\rm{x}}_0 \][/tex] is not true, then [tex]\[{\rm{\delta (x - x}}_0 ) = 0\][/tex]. The delta function is non-zero or zero as [tex]\[x \to {\rm{x}}_0 \][/tex] is true or false.

Using coordinates to label facts allow us to employ the dirac delta function to mathematically represent the implication between those facts. Does this much sound right? Thanks.

One of the most natural things to do with sets is to count the number of elements it has. This can be done in order to give a relative size to subsets. So the number of elements works as a measure of a set.

You can count an element as a member of a set only if the element is included in the set. If we label the elements with arbitrary coordinates, we can describe this mathematically as,

[tex]\[\int_{\rm{A}} {{\rm{\delta (x - x}}_0 ){\rm{dx}}} \, = \,1\][/tex]

The integral is 1 if [tex]\[{\rm{x}}_{\rm{o}} \in \,\,{\rm{A}}\][/tex]. In other words, if [tex]\[{\rm{A}} \to {\rm{x}}_0 \][/tex] is true, then the integral is 1. But if [tex]\[{\rm{A}} \to {\rm{x}}_0 \][/tex] is not true, then the integral is 0. The integral is 1 or 0 as the implication is true or false. The integrand is the dirac delta "function".

Now it seems to me that the above is all true no matter what the size of the region A is. We can let A become smaller and smaller and shrink down to the size of another point, x, and the above is still true. This procedure would be the same as taking the derivative of an integral, which is the same as the integrand, [tex]\[{{\rm{\delta (x - x}}_0 )}\][/tex]. If [tex]\[x \to {\rm{x}}_0 \][/tex] is true then [tex]\[{\rm{\delta (x - x}}_0 ) \ne 0\][/tex]. And if [tex]\[x \to {\rm{x}}_0 \][/tex] is not true, then [tex]\[{\rm{\delta (x - x}}_0 ) = 0\][/tex]. The delta function is non-zero or zero as [tex]\[x \to {\rm{x}}_0 \][/tex] is true or false.

Using coordinates to label facts allow us to employ the dirac delta function to mathematically represent the implication between those facts. Does this much sound right? Thanks.

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