Equivalence of sets proof assistance

In summary, the conversation discusses using three functions, f, g, and h, to prove that A is approximately equal to B and C. The conversation also addresses the legality of setting t=g\circf and the use of transitivity to show A<=C.
  • #1
Agent M27
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0

Homework Statement


Suppose there exist three functions:
f:A[tex]\stackrel{1-1}{\rightarrow}[/tex]B

g:B[tex]\stackrel{1-1}{\rightarrow}[/tex]C

h:C[tex]\stackrel{1-1}{\rightarrow}[/tex]A

Prove A[tex]\approx[/tex]B[tex]\approx[/tex]C

Do not assume the functions map onto their codomains.

Homework Equations


The Attempt at a Solution


I took a screenshot of my work and have attached it to the post. My question is concerning my last step, setting t=g[tex]\circ[/tex]f. Mathematically speaking I don't know if it is legal, however I don't see why I would not be able to do such an operation. Thanks in advance for the assistance.

Joe
 

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  • #2
That's fine. But there's also no reason you can't just use transitivity to show A<=C just like you did for B without using the composition.
 
  • #3
I didn't even think about the transitivity of A and C until you pointed it out! I always over complicate these things. I think I will employ that method tomorrow should this problem appear on my exam. Thanks for your help.

Joe
 

Related to Equivalence of sets proof assistance

1. What does it mean for two sets to be equivalent?

Two sets are equivalent if they have the same number of elements, denoted as |A| = |B|. This means that there is a one-to-one correspondence between the elements of the two sets. In other words, every element in set A can be paired with a unique element in set B, and vice versa.

2. What is the importance of proving the equivalence of sets?

Proving the equivalence of sets is important in various areas of mathematics, such as in algebra, calculus, and geometry. It allows us to make direct comparisons between different sets and shows that they have the same cardinality. This is crucial in understanding the relationships between different mathematical concepts and solving problems involving sets.

3. How do you prove the equivalence of two sets?

To prove that two sets A and B are equivalent, you must show that there is a bijection (a one-to-one and onto function) between the two sets. This means that for every element in set A, there is a corresponding and unique element in set B, and vice versa. You can also show the equivalence of sets by constructing a one-to-one correspondence table or using the formal definition of equivalence, which involves proving the existence of an isomorphism between the two sets.

4. Can two infinite sets be equivalent?

Yes, two infinite sets can be equivalent as long as there is a one-to-one correspondence between their elements. For example, the sets of natural numbers (N) and even numbers (2N) are both infinite sets, but they are equivalent since every natural number has a unique even number counterpart, and vice versa. This shows that the concept of equivalence of sets extends beyond finite sets.

5. Are equivalent sets the same as equal sets?

No, equivalent sets and equal sets are not the same. Two sets are equal if they have the same elements, while two sets are equivalent if they have the same cardinality. This means that two equivalent sets can have different elements, but they have the same number of elements. For example, the sets {1,2,3} and {4,5,6} are not equal, but they are equivalent since they both have three elements.

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