preprekomW

2021-08-15

Example: Partitions of Sets
a. Let $A=\left\{1,2,3,4,5,6\right\},{A}_{1}=\left\{1,2\right\},{A}_{2}=\left\{3,4\right\}$ and ${A}_{3}=\left\{5,6\right\}$. Is $\left\{{A}_{1},{A}_{2},{A}_{3}\right\}$ a partition of A?
b. Let Z be the set of all integers and let:

, and

Is $\left\{{T}_{0},{T}_{1},{T}_{2}\right\}$ a partition of Z?

AGRFTr

Step 1
a) It is known that the collection of disjoints subset of a given set or if the union of the subsets must be equal to the original set then it is called partition of sets.
Here ${A}_{1}=\left\{1,2\right\},{A}_{2}=\left\{3,4\right\},{A}_{3}=\left\{5,6\right\}$.
Find the union of the sets as follows.
${A}_{1}\bigcup {A}_{2}=\left\{1,2,3,4\right\}$ and ${A}_{2}\bigcup {A}_{3}=\left\{3,4,5,6\right\}$
Find the union of all A as follows.
${A}_{1}\bigcup {A}_{2}\bigcup {A}_{3}=\left\{1,2,3,4,5,6\right\}$.
Also ${A}_{1}\bigcap {A}_{2}=\varphi ,{A}_{2}\bigcap {A}_{3}=\varphi$ and ${A}_{1}\bigcap {A}_{3}=\varphi$.
Thus, the collection of sets $\left\{{A}_{1},{A}_{2},{A}_{3}\right\}$ are the partition of A.
Step 2
b) Here ${T}_{0}=\left\{n\in Z\mid n=3k\right\},{T}_{1}=\left\{n\in Z\mid n=3k+1\right\}$ and ${T}_{2}=\left\{n\in Z\mid n=3k+2\right\}$.
Where k is the integer.
On Substituting any integer in ${T}_{0}=\left\{n\in Z\mid n=3k\right\}$, we get ${T}_{0}=\left\{\dots -3,0,3,\dots \right\},{T}_{1}=\left\{\dots -2,1,4,\dots \right\}$ and ${T}_{2}=\left\{\dots 1,2,5,\dots \right\}$.
Take the union of the all sets as follows.
${T}_{0}\bigcup {T}_{1}\bigcup {T}_{2}=\left\{\dots -3,0,3,..\right\}\bigcup \left\{\dots -2,1,4,\dots \right\}\bigcup \left\{\dots 1,2,5,\dots \right\}$
$=\left\{\dots -2,-1,0,1,2,..\right\}$
$=Z$
Thus, the collection of set $\left\{{T}_{0},{T}_{1},{T}_{2}\right\}$ are the partition of Z.

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