Partition Axioms for Set P: Is P a Partition of Set A?

In summary, the given sets A and P satisfy the definition of a partition, as all subsets are non-empty, the union of all subsets covers the original set, and there is no overlap among the subsets. Therefore, P is indeed a partition of A.
  • #1
iHeartof12
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For the given set A, determine whether P is a partition of A.

A= ℝ, P=(-∞,-1)[itex]\cup[/itex][-1,1][itex]\cup[/itex](1,∞)

Is it correct to say that P is not partition? I don't understand why.

Thank you
 
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  • #2
A partition, as far as I know, is just a division of the set into non-intersecting subsets s.t. the union of all subsets is the original set and all subsets are non-empty.

How can you show whether or not the sets are non-intersecting? How can you show whether or not the union of the sets covers the original set? It is pretty easy to show no subset is empty.

More simply, do any of the subsets overlap with other subsets? Is there any element of ℝ that isn't in one of the subsets? If either of these is true, then the definition of a partition fails. If both are false, then we have satisfied the definition of a partition.
 
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  • #3
Here's the definition of partition as described by my one of my professor's notes:

Let [itex]X \not= \emptyset[/itex] and let each [itex]A_\alpha[/itex], where [itex]\alpha \in \Omega[/itex], be a subset of [itex]X[/itex]. Then the family of subsets [itex]\{ A_\alpha : \alpha \in \Omega \}[/itex] of [itex]X[/itex] is a partition of [itex]X[/itex] if and only if

(i) [itex]A_\alpha \not= \emptyset, \forall \alpha \in \Omega[/itex]

(ii) [itex]\bigcup_{\alpha \in \Omega} A_\alpha = X[/itex]

(iii) [itex]\forall \alpha, \beta \in \Omega[/itex], either [itex]A_\alpha = A_\beta[/itex] or [itex]A_\alpha \cap A_\beta = \emptyset[/itex].​

Does this make sense?
 
  • #4
The definition makes sense, yes. Which of these three axioms would you say are true in the problem you're considering?
 

1. What is a partition?

A partition is a collection of non-empty subsets of a set A, where each element of A is contained in exactly one subset. In other words, the subsets cover the entire set A without overlap.

2. How do you determine if P is a partition of set A?

To determine if P is a partition of set A, you need to check two conditions:

  • Each subset in P must be non-empty
  • The union of all subsets in P must equal the set A
If both conditions are satisfied, then P is a partition of set A.

3. Can a set have more than one partition?

Yes, a set can have multiple partitions. For example, the set {1, 2, 3} can have two different partitions: {{1, 2}, {3}} and {{1}, {2}, {3}}. Both of these partitions satisfy the conditions of a partition.

4. What is the difference between a partition and a subset?

A partition is a collection of subsets that covers the entire set without overlap, whereas a subset is a collection of elements from a set. A partition divides a set into non-overlapping subsets, while a subset is a smaller collection of elements within a set.

5. Can a single element be a partition of a set?

No, a single element cannot be a partition of a set. A partition must contain at least two subsets to cover the entire set without overlap. However, a singleton set with a single element can still be considered a subset of a larger set.

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