Proving Powerset P and Intersection N: A Midterm Study Guide

  • Context: Undergrad 
  • Thread starter Thread starter Klion
  • Start date Start date
  • Tags Tags
    Proof
Click For Summary
SUMMARY

The discussion focuses on proving the relationship between the powerset of the intersection of two sets A and B, specifically that P(A ∩ B) = P(A) ∩ P(B). Participants emphasize the necessity of applying definitions from set theory to establish this proof. Key steps include demonstrating that if x is in P(A ∩ B), then x must also belong to both P(A) and P(B), and vice versa. The conversation highlights the importance of understanding the definitions of powersets and subsets in order to successfully complete the proof.

PREREQUISITES
  • Understanding of set theory concepts, particularly powersets and intersections.
  • Familiarity with the notation and definitions related to subsets.
  • Basic proof techniques in mathematics.
  • Knowledge of logical equivalences in mathematical statements.
NEXT STEPS
  • Study the definitions and properties of powersets in detail.
  • Learn about subset relations and their implications in set theory.
  • Practice constructing proofs involving set operations and their properties.
  • Explore logical equivalences and their applications in mathematical proofs.
USEFUL FOR

Students preparing for midterm exams in mathematics, particularly those studying set theory, as well as educators looking for effective ways to explain the concepts of powersets and intersections.

Klion
Messages
14
Reaction score
0
P for powerset, n for intersection

show that P(AnB)=P(A) n P(B)

Studying for a midterm, seen this question in our textbook and on an old midterm. No idea how to do it. Anyone know?
 
Last edited:
Physics news on Phys.org
It's pretty basic, just apply definitions, and you should be ok.
Just prove that if x is in P(A ∩ B) then x is also in P(A) ∩ P(B), and
that if x is in P(A) ∩ P(B) then x is in P(A ∩ B).

It should just be applying definitions.
 
I know the idea behind doing a proof heh, I think my textbook is somewhat lacking though. What definitions shoudl I be attempting to make use of. Only info I've found on powersets in textbook is what the powerset actually is (the set containing all the subsets). Cant think of any helpful way to apply that to a general case though.
 
Ok:

x is in P(A ∩ B) is equivalent to saying that
x is a subset of A ∩ B
so each element χ of x is in A ∩ B
so each element χ of x is in A and in B
so x is a subset of A and x is a subset of B
so x is in P(A) and x is in P(B)

you should have no problem filling in the holes, and going in the other direction from there.
 
Shouldn't that be x is an element of...
 
Shouldn't that be x is an element of...

No, x is a subset of A∩B is correct.

x is an element of P(A∩B) which is the collection of all subsets of A∩B.
 
I see, set theory is quite new to me, it was taken off our curriclum at school and isn't much used in undergraduate physics (well not in the first year anyway).
 

Similar threads

Undergrad Prove that the tail of this distribution goes to zero
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Ways to put n balls in m boxes such that exactly k boxes are empty?
  • · Replies 29 ·
Replies
29
Views
4K
Undergrad An easy proof of Gödel's first incompleteness theorem?
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Cardinality of decreasing functions from N to N
  • · Replies 19 ·
Replies
19
Views
4K
Graduate Missing step in standard proof of |P(N)|=|R|
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Translate compound proposition p → q (implication) to p↓q question
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Question with intersects and complements.
  • · Replies 3 ·
Replies
3
Views
2K
Graduate The Current State of Mathematical Rigour
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
  • · Replies 48 ·
2
Replies
48
Views
5K