Example: Partitions of Setsa. Let A=\{1,2,3,4,5,6\}, A_{1}=\{1,2\}, A_{2}=\{3,4\}Z

preprekomW 2021-08-15 Answered

Example: Partitions of Sets
a. Let \(\displaystyle{A}={\left\lbrace{1},{2},{3},{4},{5},{6}\right\rbrace},{A}_{{{1}}}={\left\lbrace{1},{2}\right\rbrace},{A}_{{{2}}}={\left\lbrace{3},{4}\right\rbrace}\) and \(\displaystyle{A}_{{{3}}}={\left\lbrace{5},{6}\right\rbrace}\). Is \(\displaystyle{\left\lbrace{A}_{{{1}}},{A}_{{{2}}},{A}_{{{3}}}\right\rbrace}\) a partition of A?
b. Let Z be the set of all integers and let:
\(\displaystyle{T}_{{{0}}}={\left\lbrace{n}\in{Z}{\mid}{n}={3}{k},\text{for some integer}\ {k}\right\rbrace}\)
\(\displaystyle{T}_{{{1}}}={\left\lbrace{n}\in{Z}{\mid}{n}={3}{k}+{1},\text{for some integer}\ {k}\right\rbrace}\), and
\(\displaystyle{T}_{{{2}}}={\left\lbrace{n}\in{Z}{\mid}{n}={3}{k}+{2},\text{for some integer}\ {k}\right\rbrace}\)
Is \(\displaystyle{\left\lbrace{T}_{{{0}}},{T}_{{{1}}},{T}_{{{2}}}\right\rbrace}\) a partition of Z?

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Expert Answer

AGRFTr
Answered 2021-08-16 Author has 3304 answers
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 \(\displaystyle{A}_{{{1}}}={\left\lbrace{1},{2}\right\rbrace},{A}_{{{2}}}={\left\lbrace{3},{4}\right\rbrace},{A}_{{{3}}}={\left\lbrace{5},{6}\right\rbrace}\).
Find the union of the sets as follows.
\(\displaystyle{A}_{{{1}}}\bigcup{A}_{{{2}}}={\left\lbrace{1},{2},{3},{4}\right\rbrace}\) and \(\displaystyle{A}_{{{2}}}\bigcup{A}_{{{3}}}={\left\lbrace{3},{4},{5},{6}\right\rbrace}\)
Find the union of all A as follows.
\(\displaystyle{A}_{{{1}}}\bigcup{A}_{{{2}}}\bigcup{A}_{{{3}}}={\left\lbrace{1},{2},{3},{4},{5},{6}\right\rbrace}\).
Also \(\displaystyle{A}_{{{1}}}\bigcap{A}_{{{2}}}=\phi,{A}_{{{2}}}\bigcap{A}_{{{3}}}=\phi\) and \(\displaystyle{A}_{{{1}}}\bigcap{A}_{{{3}}}=\phi\).
Thus, the collection of sets \(\displaystyle{\left\lbrace{A}_{{{1}}},{A}_{{{2}}},{A}_{{{3}}}\right\rbrace}\) are the partition of A.
Step 2
b) Here \(\displaystyle{T}_{{{0}}}={\left\lbrace{n}\in{Z}{\mid}{n}={3}{k}\right\rbrace},{T}_{{{1}}}={\left\lbrace{n}\in{Z}{\mid}{n}={3}{k}+{1}\right\rbrace}\) and \(\displaystyle{T}_{{{2}}}={\left\lbrace{n}\in{Z}{\mid}{n}={3}{k}+{2}\right\rbrace}\).
Where k is the integer.
On Substituting any integer in \(\displaystyle{T}_{{{0}}}={\left\lbrace{n}\in{Z}{\mid}{n}={3}{k}\right\rbrace}\), we get \(\displaystyle{T}_{{{0}}}={\left\lbrace\ldots-{3},{0},{3},\ldots\right\rbrace},{T}_{{{1}}}={\left\lbrace\ldots-{2},{1},{4},\ldots\right\rbrace}\) and \(\displaystyle{T}_{{{2}}}={\left\lbrace\ldots{1},{2},{5},\ldots\right\rbrace}\).
Take the union of the all sets as follows.
\(\displaystyle{T}_{{{0}}}\bigcup{T}_{{{1}}}\bigcup{T}_{{{2}}}={\left\lbrace\ldots-{3},{0},{3},..\right\rbrace}\bigcup{\left\lbrace\ldots-{2},{1},{4},\ldots\right\rbrace}\bigcup{\left\lbrace\ldots{1},{2},{5},\ldots\right\rbrace}\)
\(\displaystyle={\left\lbrace\ldots-{2},-{1},{0},{1},{2},..\right\rbrace}\)
\(\displaystyle={Z}\)
Thus, the collection of set \(\displaystyle{\left\lbrace{T}_{{{0}}},{T}_{{{1}}},{T}_{{{2}}}\right\rbrace}\) are the partition of Z.
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