Cantor's Theorem (real analysis)

In summary, the conversation discusses creating a function and set for a particular homework question, and determining the 1-1 mapping from set A to P(A). The conversation also includes a link to a resource for further help on the topic.
  • #1
KevinL
37
0
Google has my particular homework online. I am doing 1.5.6, 1.5.7, 1.5.8

On 1.5.6 a), I created a function f(x) such that {a} if x = a, {b} if x = b, {c} if x=c. This is 1-1 since each element of A gets mapped to something different. Its obviously not onto.

Skipping down to 1.5.7, I need to create the set B for this function. For any a, it can't be an element of f(a). Does this mean B = the empty set?

If so, I think I can tackle the rest...
 
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  • #2
If you want help, then I suggest you post your question here. We're not going to do an in depth google search to find your homework.
 
  • #4
In this case B is going to be the empty set
 

FAQ: Cantor's Theorem (real analysis)

What is Cantor's Theorem in real analysis?

Cantor's Theorem, also known as Cantor's diagonal argument, is a fundamental result in real analysis that states that the cardinality of the real numbers is strictly greater than the cardinality of the natural numbers.

Who is Georg Cantor?

Georg Cantor was a German mathematician who is best known for his contributions to set theory, including the development of the concept of cardinality and the proof of Cantor's Theorem.

What is the significance of Cantor's Theorem?

Cantor's Theorem is significant because it provides a rigorous proof that there are different sizes of infinity, challenging the idea that all infinite sets are equivalent in size. It also has important implications in various fields of mathematics, such as topology and measure theory.

What are the applications of Cantor's Theorem?

Cantor's Theorem has various applications in mathematics, including the proof that the set of real numbers is uncountable, the development of the concept of a perfect set, and the proof of the existence of transcendental numbers.

Is Cantor's Theorem still relevant today?

Yes, Cantor's Theorem is still relevant and is widely taught and used in modern mathematics. It has also sparked ongoing debates and investigations into the nature of infinity and the concept of countability.

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