Math Problem Statement
can you show how to prove that A = {(y,x)|x is in the real numbers, y = x +1} and B = {(y,x)|y is in the real numbers, x = y - 1}
Solution
We are given two sets:
To prove these two sets are equal, we need to show that:
- Every element in is also in (i.e., ).
- Every element in is also in (i.e., ).
Proof of
Take any arbitrary element . By definition of , we know that:
- , and
- .
Thus, satisfies the condition .
Now, check whether this element satisfies the condition for being in . From the definition of , an element belongs to if .
Since , rearrange this as:
This matches the condition from the definition of , so we conclude that .
Therefore, every element of is also in , i.e., .
Proof of
Next, take any arbitrary element . By the definition of , we know that:
- , and
- .
Now, check whether this element satisfies the condition for being in . From the definition of , an element belongs to if .
Since , rearrange this as:
This matches the condition from the definition of , so we conclude that .
Therefore, every element of is also in , i.e., .
Conclusion
Since we've shown both and , we can conclude that:
Let me know if you'd like more details or clarifications!
Here are some related questions for further exploration:
- How do set-builder notations like work in set theory?
- What is the formal definition of a function and how does it relate to sets like and ?
- Can we represent these sets using other types of mathematical notation or structures?
- How do we prove that two functions are equal?
- What happens if we change the relations in sets and , like making in ?
Tip: When proving set equality, focus on showing both inclusions (i.e., and ) rather than trying to directly equate the sets. This approach is a general method for proving set equality.
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Math Problem Analysis
Mathematical Concepts
Set Theory
Set Equality
Set-Builder Notation
Formulas
y = x + 1
x = y - 1
Theorems
Set Inclusion
Set Equality
Suitable Grade Level
Grades 11-12