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:

  • A={(y,x)xR,y=x+1}A = \{ (y, x) \mid x \in \mathbb{R}, y = x + 1 \}
  • B={(y,x)yR,x=y1}B = \{ (y, x) \mid y \in \mathbb{R}, x = y - 1 \}

To prove these two sets are equal, we need to show that:

  1. Every element in AA is also in BB (i.e., ABA \subseteq B).
  2. Every element in BB is also in AA (i.e., BAB \subseteq A).

Proof of ABA \subseteq B

Take any arbitrary element (y1,x1)A(y_1, x_1) \in A. By definition of AA, we know that:

  • x1Rx_1 \in \mathbb{R}, and
  • y1=x1+1y_1 = x_1 + 1.

Thus, (y1,x1)(y_1, x_1) satisfies the condition y1=x1+1y_1 = x_1 + 1.

Now, check whether this element satisfies the condition for being in BB. From the definition of BB, an element (y,x)(y, x) belongs to BB if x=y1x = y - 1.

Since y1=x1+1y_1 = x_1 + 1, rearrange this as:

y11=x1.y_1 - 1 = x_1.

This matches the condition x=y1x = y - 1 from the definition of BB, so we conclude that (y1,x1)B(y_1, x_1) \in B.

Therefore, every element of AA is also in BB, i.e., ABA \subseteq B.

Proof of BAB \subseteq A

Next, take any arbitrary element (y2,x2)B(y_2, x_2) \in B. By the definition of BB, we know that:

  • y2Ry_2 \in \mathbb{R}, and
  • x2=y21x_2 = y_2 - 1.

Now, check whether this element satisfies the condition for being in AA. From the definition of AA, an element (y,x)(y, x) belongs to AA if y=x+1y = x + 1.

Since x2=y21x_2 = y_2 - 1, rearrange this as:

y2=x2+1.y_2 = x_2 + 1.

This matches the condition y=x+1y = x + 1 from the definition of AA, so we conclude that (y2,x2)A(y_2, x_2) \in A.

Therefore, every element of BB is also in AA, i.e., BAB \subseteq A.

Conclusion

Since we've shown both ABA \subseteq B and BAB \subseteq A, we can conclude that:

A=B.A = B.

Let me know if you'd like more details or clarifications!

Here are some related questions for further exploration:

  1. How do set-builder notations like {(y,x)condition}\{(y, x) \mid \text{condition}\} work in set theory?
  2. What is the formal definition of a function and how does it relate to sets like AA and BB?
  3. Can we represent these sets using other types of mathematical notation or structures?
  4. How do we prove that two functions are equal?
  5. What happens if we change the relations in sets AA and BB, like making y=x+2y = x + 2 in AA?

Tip: When proving set equality, focus on showing both inclusions (i.e., ABA \subseteq B and BAB \subseteq A) 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