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

preprekomW

preprekomW

Answered question

2021-08-15

Example: Partitions of Sets
a. Let A={1,2,3,4,5,6},A1={1,2},A2={3,4} and A3={5,6}. Is {A1,A2,A3} a partition of A?
b. Let Z be the set of all integers and let:
T0={nZn=3k,for some integer k}
T1={nZn=3k+1,for some integer k}, and
T2={nZn=3k+2,for some integer k}
Is {T0,T1,T2} a partition of Z?

Answer & Explanation

AGRFTr

AGRFTr

Skilled2021-08-16Added 95 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 A1={1,2},A2={3,4},A3={5,6}.
Find the union of the sets as follows.
A1A2={1,2,3,4} and A2A3={3,4,5,6}
Find the union of all A as follows.
A1A2A3={1,2,3,4,5,6}.
Also A1A2=ϕ,A2A3=ϕ and A1A3=ϕ.
Thus, the collection of sets {A1,A2,A3} are the partition of A.
Step 2
b) Here T0={nZn=3k},T1={nZn=3k+1} and T2={nZn=3k+2}.
Where k is the integer.
On Substituting any integer in T0={nZn=3k}, we get T0={3,0,3,},T1={2,1,4,} and T2={1,2,5,}.
Take the union of the all sets as follows.
T0T1T2={3,0,3,..}{2,1,4,}{1,2,5,}
={2,1,0,1,2,..}
=Z
Thus, the collection of set {T0,T1,T2} are the partition of Z.

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