Set theory

[love/hatewithMath]

Why is the following set, U, an empty set?

U = {r ∈ ℤ I 2 ≤ r ≤ -2}
​

My guess is that U is an empty set because of the ordering of the statement, with 2 and -2 written in a sequence opposite to the norm. But I don't see why that should matter. To my thinking, the statement still appears to represent a set which includes all integers, excluding those in the closed interval from -2 to 2.

 

[Deleted member 4993]

Why is the following set, U, an empty set?

U = {r ∈ ℤ I 2 ≤ r ≤ -2}
​

My guess is that U is an empty set because of the ordering of the statement, with 2 and -2 written in a sequence opposite to the norm. But I don't see why that should matter. To my thinking, the statement still appears to represent a set which includes all integers, excluding those in the closed interval from -2 to 2.

Can you provide us with

(a number) < -2​

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520​

Please share your work/thoughts about this problem.

 

[love/hatewithMath]

Can you provide us with

(a number) < -2

Yes. I give you, in answer, the number -3. Are there any more tricks you'd like for me to perform?

Please share your work/thoughts about this problem.

Since my attempt to explain my thoughts verbally in the original post, by writing "To my THINKING..." have failed, I'll try to represent the way that I interpret the original statement graphically. First, I'll repost the original statement for which I requested assistance:

U = {r ∈ ℤ I 2 ≤ r ≤ -2}

Now, below you will find my attempt to represent the same statement graphically:



Finally, I'll restate my original question:

Why is it that the written set definition is judged to be an empty set, whereas one who may not understand the syntax (maybe me) might interpret the definition graphically as I have, which depicts a set containing all integers, excluding those in the closed interval (-2,2)?

 

[Deleted member 4993]

Yes. I give you, in answer, the number -3. Are there any more tricks you'd like for me to perform?


Since my attempt to explain my thoughts verbally in the original post, by writing "To my THINKING..." have failed, I'll try to represent the way that I interpret the original statement graphically. First, I'll repost the original statement for which I requested assistance:

Now, below you will find my attempt to represent the same statement graphically:

View attachment 21872

Finally, I'll restate my original question:

Why is it that the written set definition is judged to be an empty set, whereas one who may not understand the syntax (maybe me) might interpret the definition graphically as I have, which depicts a set containing all integers, excluding those in the closed interval (-2,2)?

Think about difference between "AND" & "OR"

If the statement was:

-2 <= r <=2 ..................................................................... (A)

then all the numbers between -2⁺ and +2^- populate the set. We have - for example -1

-2 < -1 .................AND.............. -1 < 2

However the statement is

-2 => r =>2 ..................................................................... (B)

here we cannot find an equivalent statement. Any number cannot be less than -2 AND be greater than +2. Thus the statement (B) represents an empty set.

The graphic above represents numbers less than -2 OR greater than +2

 

[pka]

The statement that \(2\le x\le -2\) is read is "\(x\) at least two and at most minus two". Is there any such number?

 

[Steven G]

x>2 and x<-2. No such number.

2< x <-2 means that 2 < -2 which is not true!

 

[love/hatewithMath]

I see now. Thanks to all. My blind spot was in the way I read the statement 2 ≤ r ≤ -2. I did not realize that the statement translates to "r at least two AND at most minus two".

Could the math statement be rewritten to indicate "r at least two OR at most minus two"? If so, what does that look like?

 

[Dr.Peterson]

I see now. Thanks to all. My blind spot was in the way I read the statement 2 ≤ r ≤ -2. I did not realize that the statement translates to "r at least two AND at most minus two".

Could the math statement be rewritten to indicate "r at least two OR at most minus two"? If so, what does that look like?

You would just use the word "or" (or, if you have learned it, a symbol for "or", namely "[MATH]\vee[/MATH]"):

U = {r ∈ ℤ I r ≥ 2 or r ≤ -2}​

It can also be expressed as the union of two sets:

U = {r ∈ ℤ I r ≥ 2} [MATH]\cup[/MATH] {r ∈ ℤ I r ≤ -2}​