Don't understand why this set is bounded

Hey all,

We were discussing bounded and unbounded sets in class, and looking over my notes, I see that I have some trouble understanding the concept.

Here are three examples that our professor gave us:

Set A = {x$$\in$$R | |x| <10}
Set A = {x$$\in$$R | x<10}

A$$\subseteq$$Z s.t. x~y iff x|y
Set A = {1,2,3,4,5,8}

Supposedly the first one is bounded, the second one is not, and the third one has a lower bound of 1 but does not have an upper bound.

I just genuinely don't understand why this is the case...anyone care to explain?

Thanks so much!

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First of all you need a precise mathematical definition of a bounded set. Without such a definition you will be lost forever. Do you have such a definition in your notes?

HallsofIvy
Homework Helper
Hey all,

We were discussing bounded and unbounded sets in class, and looking over my notes, I see that I have some trouble understanding the concept.

Here are three examples that our professor gave us:

Set A = {x$$\in$$R | |x| <10}
Set A = {x$$\in$$R | x<10}

A$$\subseteq$$Z s.t. x~y iff x|y
This defines an equivalence relation. Do you mean the set of ordered pairs of integers, (x, y), such that x divides y? If so, how are you defining the "distance"?

Set A = {1,2,3,4,5,8}

Supposedly the first one is bounded, the second one is not, and the third one has a lower bound of 1 but does not have an upper bound.

I just genuinely don't understand why this is the case...anyone care to explain?

Thanks so much!
I second Arkajad's suggestion- write out precisely what the definitions of "bounded", "bounded below", and "bounded above" are!

Here are three examples that our professor gave us:

Set A = {x$$\in$$R | |x| <10}
The absolute value of x must be less than 10 meaning it must be in the interval (-10, 10).

Set A = {x$$\in$$R | x<10}
This one isn't bounded below since x can become arbitrarily small by becoming more negative. For example x would be in the interval (−∞, 10)
A$$\subseteq$$Z s.t. x~y iff x|y
Set A = {1,2,3,4,5,8}
By x|y do you mean x as a factor of y?