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The collections are generally linked in a given context. If we think of ourselves, then we belong to a certain society, which in turn belongs to a province, which in turn belongs to a country and so on. In the context of a school, all students of a school belong to school. Some of them belong to a certain class. If there are sections within a class, then some of these belong to a section.
We need to have a mathematical relationship between different collections of similar types. In set theory, we denote this relationship with the concept of “subset”.
We use symbol “ $\subset $ ” to denote this relationship between a “subset” and a “set”. Hence,
$$A\subset B$$
We read this symbolic representation as : set “A” is a subset of set “B”. We express the intent of relationship as :
$$A\subset B\phantom{\rule{1em}{0ex}}\mathrm{if}\phantom{\rule{1em}{0ex}}x\in A,\phantom{\rule{1em}{0ex}}\mathrm{then}\phantom{\rule{1em}{0ex}}x\in B$$
It is evident that set "B" is larger of the two sets. This is sometimes emphasized by calling set "B" as the “superset” of "A". We use the symbol “ $\supset $ ” to denote this relation :
$$B\supset A$$
If set "A" is not a subset of "B", then we write this symbolically as :
$$A\not\subset B$$
Some of the important characteristics and related deductions are presented here :
It is clear that set “B” is inclusive of subset “A”. It means that “B” may have additional elements over and above those common with “B”. In case, all elements of “B” are also in “A”, then two sets are equal. We express this symbolically as :
$$If\phantom{\rule{1em}{0ex}}A\subset B\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}B\subset A,\phantom{\rule{1em}{0ex}}\mathrm{then}\phantom{\rule{1em}{0ex}}A=B.$$
This is true in other direction as well :
$$If\phantom{\rule{1em}{0ex}}A=B,\phantom{\rule{1em}{0ex}}\mathrm{then}\phantom{\rule{1em}{0ex}}A\subset B\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}B\subset A.$$
We can write two instances in a single representation as :
$$A\subset B\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}B\subset A\iff A=B$$
The symbol “ $\iff $ ” means that relation holds in either direction.
Every set is subset of itself. This is so because every element is present in itself.
Empty set is a subset of every set. This deduction is direct consequence of the fact that empty set has no element. As such, this set is subset of all sets.
We have seen from the deductions above that special circumstance of “equality” can blur the distinction between “set” and “subset”. In order to emphasize, mother-child relation between sets, we coin the term “proper subset”. If every element of set “B” is not present in set “A”, then “A” is a “proper” subset of set “B”; otherwise not. This means that set “B” is a larger set, which, besides other elements, also includes all elements of set “A”.
Set of vowels in English alphabet, “V”, is a “proper” subset of set of English alphabet, “E”. All elements of “V” are present in “E”, but not all elements of “E” are present in “V”.
There is a bit of conventional differences. Some write a “proper” subset relation using symbol “ $\subset $ ” and write symbol “ $\subseteq $ ” to mean possibility of equality as well. We have chosen not to differentiate two subset types.
The number system is one such system, in which different number groups are related. Natural number is a subset of integers. integers are subset of rational numbers and rational numbers are subset of real numbers. None of these sets are equal. Hence, relations are described by proper subsets.
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