Equivalence between a set and the subset of its subset

In summary, equivalence between a set and the subset of its subset is a mathematical concept that describes the relationship between a set and its subset. It is a fundamental concept in set theory that helps establish the hierarchical structure of sets and their subsets. Examples of this concept include the relationship between sets of even numbers and prime numbers. It is important in mathematics as it allows for the comparison and classification of sets, and has practical applications in computer science and data analysis.
  • #1
batballbat
127
0
Is it true that: If A is not equivalent to its subset A1. Then A is not equivalent to any subset of A1?
 
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  • #2
You should consider also informing us about the relevant equivalence relation, as the truth or falsity of your statement heavily depends on that information.

For example, if you for your eq.rel. use existence of a bijection, the statement is true, but other relations, like having same parity, will render your statement false.
 
  • #3
actually this is trivial. I just learned all this the complicated way, .i.e. proving cantor bernstein without the well ordering theorem.
 

1. What is the definition of equivalence between a set and the subset of its subset?

Equivalence between a set and the subset of its subset is a mathematical concept that refers to the relationship between a set and its subset. It states that if a set A is a subset of a set B, and a subset C of A is also a subset of B, then A and C are equivalent.

2. How does equivalence between a set and the subset of its subset relate to set theory?

Equivalence between a set and the subset of its subset is a fundamental concept in set theory. It helps to establish the hierarchical structure of sets and their subsets, and allows for the comparison and classification of sets based on their relationships.

3. What are some examples of equivalence between a set and the subset of its subset?

One example of equivalence between a set and the subset of its subset is the relationship between the set of even numbers and the subset of even square numbers. Another example is the relationship between the set of prime numbers and the subset of prime numbers greater than 10.

4. Why is equivalence between a set and the subset of its subset important in mathematics?

Equivalence between a set and the subset of its subset is important in mathematics because it helps to establish the structure and properties of sets. It also allows for the comparison and classification of sets, which is essential in many areas of mathematics including algebra, number theory, and topology.

5. How can equivalence between a set and the subset of its subset be applied in real-world situations?

Equivalence between a set and the subset of its subset has many practical applications, especially in computer science and data analysis. It can be used to categorize and organize data, identify patterns and relationships, and make predictions based on subsets of larger datasets.

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