Earlier we read about Sets . Today we will see some basic definitions of sets.
For all our definitions we will use the example of the sum of the values obtained in simultaneous rolling of two dice.
$\underline{I.\ Universal\ Set}$
Universal set defines the set of all possible elements meeting the condition. In our example of rolling of two dice the possible sum of the values in each die is $2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11,\ 12$.
Therefore, in this case the universal set $(U)$ will be:
Cardinality of a set is the count of elements in the set. This is denoted by two pipe $\unicode{0x201C}\ |\ \unicode{0x201D}$ symbols around the set name.
In the previous case, since we have $11$ elements in the set $U$, we can write:
$|U| = 11$
$\underline{III.\ Subsets}$
When all elements of a set $(A)$ are contained by another set $(B)$, then we say that $\unicode{0x201C}A\ is\ a\ subset\ of\ B\unicode{0x201D}$. This is denoted by:
$A \subset B$ or $A \subseteq B$. There is a difference between these two.
When we write
$A \subset B$ it means $A$ is a $proper\ subset$ of $B$, in other word $A$ contains less elements than $B$.
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When we write:
$A \subseteq B$ it means $A$ is an $improper\ subset$ of $B$, that is $A$ can contain either less or exactly the same elements $B$.
In our example, if we say
$Set\ B$ = all outcomes that are even, then:
$Set\ B = \left\{2, 4, 6, 8, 10, 12 \right\}$
$Set\ A$ = all outcomes that are divisible by $4$, then:
$Set\ A = \left\{4, 8, 12 \right\}$
We can write $A \subset B$
$\underline{IV.\ Mutually\ Exclusive\ Sets}$
When two set $A$ and $B$ share no common element, then they are called $Mutually\ Exclusive\ Sets$
In our example if we define:
$Set\ A$ = all outcomes that are divisible by $4$, then:
$Set\ A = \left\{4, 8, 12 \right\}$
$Set\ B$ = all outcomes that are divisible by $5$, then
$Set\ B = \left\{5, 10 \right\}$
Since $A$ and $B$ does not have any common element we can say $A$ and $B$ are mutually exclusive sets.
We denote this by:
$A \cap B = \phi$ (Remember, $\phi$ denotes null set and $\cap$ denotes intersection)
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$\underline{V.\ Exhaustive\ Sets}$
When the union of two or more sets equals the universal set, then they are exhaustive sets.
Let us see this from our example:
Let $A$ be the set of all outcomes that are perfect squares, then:
$Set\ A = \left\{4, 9 \right\}$
Let $B$ be the set of all outcomes that are primes, then:
$Set\ B = \left\{2, 3, 5, 7, 11 \right\}$
Let $B$ be the set of all outcomes that are even, then:
$Set\ C = \left\{2, 4, 6, 8, 10, 12\right\}$
Now we can see that $A \cup B \cup C = \left\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 \right\} = U$
Therefore $A$, $B$ and $C$ are exhaustive sets.
There are many more definitions used in Set Theory. For now, what we have covered should be enough.
$\underline{VI.\ Power\ Sets}$
Power set of a given set $A$, is a set of all possible subsets of $A$
If we define the set $A$ as:
$A = \{1,\ 2,\ 3\}$
then the possible subsets, including the null subset and the equal subset are: $\{\}$
Suppose a given set $A$ contains $n$ elements, the each of the elements in $A$ can be in either of the $2$ states in the subsets of $A$, that it can either be present in a subset or it can be absent in the subset. If you pick any random element of set $A$ in the above example, you will observe that each element is present in exactly $4$ subsets and absent in the other $4$ subsets.
Therefore, there can be:
$(2 \times 2 \times 2\ ...\ upto\ n\ terms)$ possible subsets of $A$, which is equal to $2^n$
Therefore, the order of the power set of $A$ is $2^n$ where $n$ is the order of $A$, and the power set will include the $null$ subset and the equal subset of $A$ as well.