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Here are some suggestions for memorizing the difference between an open interval and a closed interval.

[a,b]={x|a<=x<=b}

(a,b)={x|a<x<b}

The first thing I would notice is that the bracket (square bracket), looks a little bit like the letter

1.

|= (the letter C, but, draw a square C, otherwise you get confused. I cannot do it in ascii.

...

2.

Remember, that it's a square C

The parenthesis, is a little bit like the letter O (notice the roundness). O, is like, "omit".


To envision that the dot near the equal, part, is included, envision a door closing (this is the square bracket, with the door being the vertical edge and the doors by the door being the other two horizontal edges. The edges, close, onto the door, and include (by squishing it), the point that is in it.

|= ... |•|

Again, this is hard to draw. A video would be needed.

2.

For the open interval, envision a rubber band, like a spring. The rubber band pushes the dot on the edge, out, (onto the points not included in the endpoint).

Do things right.

Use this mnemonic from the beginning.

You will never get confused.

....

For those without textbook, let me read the notation for you.

[a,b]={x|a<=x<=b}

The closed interval a b is the set of all x, such that (| is read such that), a is less than or equal to x is less than or equal to b.

(a,b)={x|a<x<b}

The open interval a b is the set of all x, such that, a is strictly less than x is strictly less than b.

....

x is a variable

The stuff inside the { } (curly brackets), denotes a set.

a and b are points (constants on the real line).

.....

How would you make improvements to this explanation, and, more importantly, what mnemonics would you utilize to make sure the students get it right, and, then, get it right for the rest of their life.

This post is about notation, and, about, making it such, that the student does not get confused by notation.

Thanks.

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    $\begingroup$ Just a minor addition: I've also seen $]a,b[$ with the brackets facing outwards for open intervals. $\endgroup$
    – J W
    Jan 21 at 7:55
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    $\begingroup$ I don't think this is necessarily a bad question, although it is always a bit concerning when people ask for mnemonics: at some stage in your maths learning journey, you should get to the point where reading and trying to understand definitions and doing lots of exercises based on these definitions will train you enough so that you know the definitions without really memorising them. That's partly the point of doing exercises (i.e. lots of questions) on a topic. $\endgroup$ Jan 21 at 18:55

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Honestly this all feels a bit over the top to me. I think all that needs to be said is that the open interval $(a,b)$ fits inside the closed interval $[a,b].$ That's what I've always thought was the intuition behind the notation anyway. An oval $()$ fits inside a rectangle $[\,\,]$ with the same width and height.

It even looks that way visually with inequalities. Literally, the physical text $a < x < b$ fits inside the text $a \leq x \leq b.$ The latter is just the former with some extra lines underneath.

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    $\begingroup$ Interesting view, which I hadn't seen or thought of before. In my case, when I first saw this notation in class (Fall 1973 in 9th grade Algebra 1, using Dolciani's book), or earlier (which is most likely the case) from seeing it in out-of-school reading, square brackets with their straight-line segments seemed for some reason more visually connected with having endpoints than arcs of a circle. Not sure why, maybe because circles don't have endpoints? And yes, I know arcs can $\ldots$ $\endgroup$ Jan 21 at 20:13
  • $\begingroup$ @DaveLRenfro, maybe because arcs barely touch the endpoints whereas the straight-line segments fully include them? $\endgroup$
    – J W
    Jan 21 at 20:40

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