Sets: Proving Y^(XU{x})=(Y^X)x(Y^{x}) with Finite Sets and Singleton {x}

  • Context: Graduate 
  • Thread starter Thread starter MathematicalPhysicist
  • Start date Start date
  • Tags Tags
    Sets
Click For Summary

Discussion Overview

The discussion revolves around proving the equation Y^(XU{x})=(Y^X)x(Y^{x}) for finite sets X and Y, where {x} is a singleton. Participants explore methods of demonstrating this relationship, including the use of bijections and counting arguments.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant seeks guidance on how to show the containment of one set within another to prove the equation.
  • Another participant clarifies that proving |Y^X|=|Y|^|X| is the goal and mentions attempting an inductive proof.
  • A suggestion is made to write down a bijection to demonstrate the relationship, implying that direct counting could suffice.
  • There is a discussion on the nature of counting as a rigorous proof method, with one participant asserting that counting is indeed rigorous.
  • Another participant expresses skepticism about counting being rigorous enough and insists on needing a more formal proof.

Areas of Agreement / Disagreement

Participants express differing views on the adequacy of counting as a proof method, with some supporting it as rigorous while others seek a more formal approach. The discussion does not reach a consensus on the best method to prove the equation.

Contextual Notes

Participants have not resolved the assumptions or definitions related to the cardinality of sets and the nature of bijections in this context.

MathematicalPhysicist
Science Advisor
Gold Member
Messages
4,662
Reaction score
372
how do i show that Y^(XU{x})=(Y^X)x(Y^{x}) where X and Y are finite sets, and {x} is a singleton.
obvisouly i need to show that one set is contained in another and vice versa, the problem is how to do so?
 
Last edited:
Physics news on Phys.org
for those who haven't undersatand, i need to prove |Y^X|=|Y|^|X|
i tried to prove it in induction on the exponent but i got to what i posted in the first post in this thread, can someone help me on this here?
 
Just write down a bijection, in the first post. Though the result you want to prove in the second post follows from counting the elements directly.
 
matt grime said:
Though the result you want to prove in the second post follows from counting the elements directly.
you mean because the set of all functions from X to Y, its cardinal equals the number of possible mappings from X to Y, which is |Y|^|X|, right?
still i think that i need a rigorous proof for this, and counting isn't as rigouros.
 
Of course counting is rigorous.
 

Similar threads

Undergrad Is this conditional expectation identity true?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Error bars in Excel (standard deviation, standard error, percentage)
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Thinking about equality of infinite sets
  • · Replies 5 ·
Replies
5
Views
3K
MHB Does AxA Equal BxB Imply A Equals B?
  • · Replies 5 ·
Replies
5
Views
5K
Undergrad Statistics proof: y = k x holds for a data set
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Probability that X is less than a set
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Countability of binary Strings
  • · Replies 18 ·
Replies
18
Views
4K
MHB Proving Countability: Positive Rational Numbers & Union of Countable Sets
  • · Replies 11 ·
Replies
11
Views
3K
MHB Proof of Sets X,Y: X⊆Y <=> P(X)⊆P(Y)
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Contrapositive of theorem and issue with proof
  • · Replies 5 ·
Replies
5
Views
2K