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



Answered question


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



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.
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.
Thus, the collection of set {T0,T1,T2} are the partition of Z.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Discrete math

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?