What do INCLUDING and NOT INCLUDING mean

[ThunderBalls97]

I'm doing functions in maths but don''t know what including or not including means.

 

[Deleted member 4993]

I'm doing functions in maths but don''t know what including or not including means.

Can you please give us some examples where those two terms are used - and you don't understand?

 

[Steven G]

There are many cases where mathematicians use include and not include. As Khan asked, can you please give us an example.

 

[ThunderBalls97]

Its not really an example its just the basics where you have the white dots and he black dots.

 

[Deleted member 4993]

Its not really an example its just the basics where you have the white dots and he black dots.

You donot mention INCLUDING or NOT INCLUDING in your statement above.

 

[ThunderBalls97]

Well in the textbook it talks about including and not including so what does it mean

 

[Deleted member 4993]

Well in the textbook it talks about including and not including so what does it mean

Please give a quotation/s from your textbook (indicating your confusion).

 

[ThunderBalls97]

 

[HallsofIvy]

I'm beginning to think this is just a joke. When asked for an example, the poster gave an "example", "black dots" and "white dots", that had nothing to do with "including" or "not including" and when asked about that, just repeated the first post.

 

[ThunderBalls97]

 

[ThunderBalls97]

Mate I'm not joking I'm struggling in the chapter I'm doing in class and just need to know what including or not including the square and circle brackets mean when it comes to notation.

 

[pka]

I'm doing functions in maths but don''t know what including or not including means.

I think this what you mean. I do not think it is a joke at all.

The above are the plots of four sets on a number line.
They are: $(-2,4),~[-2,4),~(-2,4],~\&~[-2,4]$.
Note the use of $<,~\le,~>~\&~\ge$ with the corresponding $\circ-\circ,~\bullet-\circ,~\bullet-\circ~\&~\bullet-\bullet$
These indicate the inclusion and or exclusion of rhe endpoints of the sets.

 

[ThunderBalls97]

So what does inclusion or exclusion if exclusion of the endpoints mean

 

[HallsofIvy]

Oh, thanks, PKA, that was my first thought but the "white dots and black dots". I have always called those "open and closed dots". I apologize if I seemed unnecessarily snarky.

The notation (0, 1) means "all numbers from 0 to 1, not including 0 and 1. 0 is NOT INCLUDED in that set which simply means there are numbers arbitrarily close to 0 (0.000001, 0.0000000....0000001) in the set but 0 itself in the set. Similarly, in [0, 1) or [0, 1], the number 0 is "included" in the set simply meaning it is in the set.

Simply put, "a IS INCLUDED in set X" means that a is in set X. "a IS NOT INCLUDED in set X" means that a is not in the set.

 

[ThunderBalls97]

So why do numbers have to be included

 

[pka]

So why do numbers have to be included

If we say that $x\ge 2$ that is read "x is at least two". Now is $2$ included or not?

If we say that $3<y$ that is read "y is greater than three". Now is $3$ included or not?

 

[ThunderBalls97]

If we say that $x\ge 2$ that is read "x is at least two". Now is $2$ included or not?

If we say that $3<y$ that is read "y is greater than three". Now is $3$ included or not?

No 3 is not included

 

[Steven G]

I am not clear what you are asking but your latest posit is more clear than your first post.

Suppose 5< x <15. So for sure x can be any number between 5 and 15. These numbers (between 5 and 15) are highlighted on the number line. Now comes the question. Should we include 5 and 15, and if so how do we show this on the number line. We CAN include 5 and NOT include 15. I hope that you can understand this. Now for the number between 5 and 15 are highlighted, FILLED IN. Now if YOU got to decide how to denote that you want to include 5 how would you do this-- with an open circle or a FILLED IN circle? How would you denote that you do not want to include 15- with an open circle or a FILLED IN circle? I bet that YOU will make the correct choice!