Proving the Same Cardinality of Sets A & B

In summary, to prove that two sets have the same cardinality, a one-to-one correspondence (or bijection) between their elements must be shown. A one-to-one correspondence is a function that maps each element of one set to a unique element in another set. It is possible for two sets to have the same cardinality even if one is a subset of the other. There are different levels of cardinality, such as finite and infinite, and proving the same cardinality of sets is significant in mathematics for comparing and classifying sets, understanding their properties and relationships, and studying set theory and other branches of mathematics.
  • #1
evinda
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MHB
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Hello! (Smile)

If we want to show that the sets:

$$A=\{ 3X^2| X \in \mathbb{Z}_p \}\ \ \text{ and } \ \ B=\{ 7-5Y^2| Y \in \mathbb{Z}_p\}$$

have the same cardinality, could we take the bijective function $f$ such that $f(x)=\frac{7-5x}{3}$ ? Or am I wrong? (Thinking)
 
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  • #2
Yes, you are correct! The function $f$ is a bijective function which maps elements of $A$ to elements of $B$ and vice versa. This proves that the two sets have the same cardinality.
 

FAQ: Proving the Same Cardinality of Sets A & B

How do you prove that two sets have the same cardinality?

To prove that two sets have the same cardinality, you must show that there exists a one-to-one correspondence (or bijection) between the elements of the two sets. This means that each element in one set is paired with exactly one element in the other set, and vice versa.

What is a one-to-one correspondence?

A one-to-one correspondence, also known as a bijection, is a function that maps each element of one set to a unique element in another set. This means that there are no repeating elements in either set, and every element in one set is paired with exactly one element in the other set.

Can two sets have the same cardinality if one set is a subset of the other?

Yes, two sets can have the same cardinality even if one set is a subset of the other. This is because the number of elements in a set does not necessarily determine its cardinality, but rather the relationship between the elements of the two sets.

Are there different levels of cardinality?

Yes, there are different levels of cardinality. The most commonly used levels are finite (countable) and infinite (uncountable), but there are also different levels of infinity within the infinite cardinality category.

What is the significance of proving the same cardinality of sets?

Proving the same cardinality of sets is important in mathematics because it allows us to compare and classify sets based on their size. It also helps us understand the properties and relationships between different sets, and is a fundamental concept in the study of set theory and other branches of mathematics.

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