MHB Is this a well-formed set-builder notation?

AI Thread Summary
Sets A and C are equivalent as they both represent ordered pairs in R², while set B is not equivalent due to its ambiguous notation, which does not specify the order of elements. The discussion highlights that A and C correctly define their elements as ordered pairs, making the order significant, whereas B lacks this clarity. A specific example illustrates that while (5,1) does not belong to A, it could satisfy the condition for B, demonstrating the difference in interpretation. Additionally, B is criticized for not adhering to well-formed set-builder notation, which should clearly define the relationship between elements. Overall, the consensus confirms that A and C are equivalent, but B is not.
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Are these three sets equivalent?

$$A=\left\{(x,y):x,y\in\Bbb{R},y\ge x^2-1\right\}$$
$$B=\left\{x,y\in\Bbb{R}:y\ge x^2-1\right\}$$
$$C=\left\{(x,y)\in\Bbb{R}^2:y\ge x^2-1\right\}$$

I am thinking that $A$ and $C$ are, but not $B$ as it might be ambigious as to which dimension it is in, i.e it could be in $\Bbb{R}^3$, where any value of $z$ will satisfy. Am I right? :D
 
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Rido12 said:
Are these three sets equivalent?

$$A=\left\{(x,y):x,y\in\Bbb{R},y\ge x^2-1\right\}$$
$$B=\left\{x,y\in\Bbb{R}:y\ge x^2-1\right\}$$
$$C=\left\{(x,y)\in\Bbb{R}^2:y\ge x^2-1\right\}$$

I am thinking that $A$ and $C$ are, but not $B$ as it might be ambigious as to which dimension it is in, i.e it could be in $\Bbb{R}^3$, where any value of $z$ will satisfy. Am I right? :D

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Yes, you are right.

At the set $A$ we have the ordered pair $(x,y)$ such that $x,y \in \mathbb{R}$ so it is meant that $(x,y) \in \mathbb{R}^2$.
The set $B$ isn't equivalent to the other two because of the fact that at the sets $A,C$ we consider an ordered pair and and so the order in which the objects appear is significant, but for the set $B$ this doesn't hold.
For example, if we are given $x=5$, $y=1$, we check if $(5,1) \in A$ that does not hold since it doesn't hold that $1 \geq 5^2-1$.
For the set $B$ we check if $y \geq x^2-1$ which holds by taking $y=5$ and $x=1$.
 
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Rido12 said:
Are these three sets equivalent?
Sets can be equal, and definitions can be equivalent.

Rido12 said:
$$B=\left\{x,y\in\Bbb{R}:y\ge x^2-1\right\}$$
This is not a well-formed set-builder notation. A well-formed notation has the form $\{x\mid P(x)\}$, $\{x\in A\mid P(x)\}$ or $\{f(x)\mid P(x)\}$, but not $\{x,y\in A\mid P(x,y)\}$.
 
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