Problem on sets, subsets: squares, rectangles, rhombuses, quadrilaterals

[chijioke]

Here are my answers:
a)true because a square is a special rectangle.

b)false a rectangle can never be rhombus.

c)true because a rhombus is a set of quadrilateral.

d)true because a square is a special rhombus.

e)true because a square is a set of quadrilateral.
{f)f a rectangle can never be a square.

g)because a rectangle is a set of quadrilateral.

h)false because every quadrilateral is not square.

i)false because a square is a special rectangle and also a rectangle belong to a set of quadrilateral.
How good are my ideas?

 

[The Highlander]

How good are my ideas?

Some are correct others are wrong.

f) has the correct answer but for the wrong reason; your thinking is faulty! (You said, at a), that: "a square is a special type of rectangle" (true) but that means that there are rectangles that are square so it wrong to then say: "a rectangle can never be a square."
(When I was in primary school our teachers taught us to call a non-square rectangle an oblong. This is a valuable distinction that has been lost nowadays and so we get kids coming up to high school believing that a square and a rectangle are two completely different things, whereas we (my generation) knew that squares and oblongs were both rectangles! Unfortunately, many of the current (primary school) teachers fail to understand this themselves; hence the persistence of the misconception! ?)

g) you have given no answer! (True or False)

I will leave it to you to go through your answers again and see if you can correct your mistakes (there are more than the just the two I have pointed out).

Hint: Draw a Venn diagram for the sets defined in the question then rework your answers. ??
(Submit your Venn diagram along with your revised answers (to support them); such a diagram will illustrate your answers without further explanation. ?)

 

[The Highlander]

I will start you off...
Download & complete the Venn diagram (below).
(A simple graphics editor like Windows Paint™ or even MS Word™ will do the job.)

What letter (A, B, C or D) goes into the big circle to identify the set it represents?
You need to add further circles and put a letter into each 'space' created to identify the set that 'space' represents.

​

Then you should find it easier to answer all of the questions (correctly). ?

 

[Dr.Peterson]

How good are my ideas?

Here are some corrections:

Here are my answers:​

a)true because a square is a special rectangle. Correct​

​

b)false a rectangle can never be rhombus. Compare your answers to (a) and (d); but this is not what they are asking​

​

c)true because a rhombus is a set of quadrilateral. Correct​

​

d)true because a square is a special rhombus. Correct​

​

e)true because a square is a set of quadrilateral. Correct​

​

f) [false?] a rectangle can never be a square. Compare your answer to (a); but this is not what they are asking​

​

g)[true?] because a rectangle is a set of quadrilateral. Correct​

​

h)false because every quadrilateral is not square. Correct; but you should say "not every quadrilateral is a square"​

​

i)false because a square is a special rectangle and also a rectangle belong to a set of quadrilateral. Why do you say false?​

(I hope I got all that right! There's a lot to check.)

When we say X is a subset of Y, we are saying that everything in X is also in Y; it will be false if there is something in X that is not in Y. It does not mean that nothing in X is in Y.

 

[stapel]

(When I was in primary school our teachers taught us to call a non-square rectangle an oblong. This is a valuable distinction that has been lost nowadays and so we get kids coming up to high school believing that a square and a rectangle are two completely different things, whereas we (my generation) knew that squares and oblongs were both rectangles!...)

Huh. Interesting. Thank you for sharing that! ?

 

[Dr.Peterson]

(When I was in primary school our teachers taught us to call a non-square rectangle an oblong. This is a valuable distinction that has been lost nowadays and so we get kids coming up to high school believing that a square and a rectangle are two completely different things, whereas we (my generation) knew that squares and oblongs were both rectangles! Unfortunately, many of the current (primary school) teachers fail to understand this themselves; hence the persistence of the misconception!)

This may not, of course, be true of all schools everywhere; but the misconception is certainly common. I just attribute it to the difference between everyday usage, which tends to be exclusive ("no, I asked for the rectangle, not the square!"), and technical usage, which tends to be inclusive ("this theorem applies to all rectangles, and therefore specifically to squares").

I recently had a discussion in which someone pointed out that Euclid's Elements, Book I, Def. 22, says, as they quoted it to me: “Of the quadrilateral figures: a square is that which is right-angled and equilateral, a rectangle that which is right-angled but not equilateral, a rhombus that which is equilateral but not right-angled”. That implies that squares are not rectangles (or rhombuses).

But then I checked my usual source for Euclid, and found that it had a better translation: "Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia."

I know enough Greek to check this, and the latter translation is indeed better. There are two different words, one of which literally means oblong ("different-sided"), and the other literally means rectangular ("right-angled"). So the first translator made the same mistake you refer to. (But ultimately it's a matter of definition; apparently the Greeks defined the rhombus exclusively, like the oblong.)

 

[The Highlander]

Here are my answers:
a)true because a square is a special rectangle.

b)false a rectangle can never be rhombus.

c)true because a rhombus is a set of quadrilateral.

d)true because a square is a special rhombus.

e)true because a square is a set of quadrilateral.
{f)f a rectangle can never be a square.

g)because a rectangle is a set of quadrilateral.

h)false because every quadrilateral is not square.

i)false because a square is a special rectangle and also a rectangle belong to a set of quadrilateral.
How good are my ideas?

Since @Dr.Peterson has given you the answers, I don't suppose you will bother to attempt what I suggested (at Posts 2 & 3) but I think it is important that you see why I suggested what I did in those posts (and it may also help other, future(?) readers to understand that too), so I have completed the Venn diagram (below) and added some pretty colours to it. ?

X \(\displaystyle \subset\) Y means that the set X is a proper subset of the set Y. Two conditions must prevail for that to be true:-

1. All the members of the set X must also be members of the set Y.​

and

2. The set Y must also contain members that are not in the set X.​

You might rephrase that (non-mathematically) as: "X must be completely inside Y but Y must be 'bigger' than X".

So you can easily see whether X \(\displaystyle \subset\) Y is True or False by drawing a Venn diagram.

Take, for example, part a) in your question. The yellow (yellow because red+green=yellow) space in the Venn diagram, labelled A (representing all squares), lies completely within the green circle, labelled B (representing all rectangles), but there is plenty of (green) space that is "left over", ie: there is "room to spare" you might say.
Therefore, (space) A is completely inside (space) B and (space) B is 'bigger' than (space) A which means that "A \(\displaystyle \subset\) B" is True. ?

​

Do you now understand how you can see at a glance whether each of the relationships given is True or False by just looking at how they appear on the diagram?

Thus...

a) A \(\displaystyle \subset\) B? True (Because A is completely inside B and B is bigger than A.)

b) B \(\displaystyle \subset\) C? False (Because B is not completely inside C.)

c) C \(\displaystyle \subset\) D? True

d) A \(\displaystyle \subset\) C? True

e) A \(\displaystyle \subset\) D? True

f) B \(\displaystyle \subset\) A? False

g) B \(\displaystyle \subset\) D? True

h) D \(\displaystyle \subset\) A? False

and

i) A \(\displaystyle \subset\) B \(\displaystyle \subset\) D? True

Furthermore, if you approach the question in this fashion (by drawing the Venn diagram and then listing your answers as I have done, above) then you do not need to offer any "explanations" for your answers because the diagram illustrates explicitly what the correct answers are, graphically.
(I just added a couple of explanations for the first two for 'clarification'.)

You got into 'trouble' trying to explain your answers, in words, on more than one occasion and adopting this alternative approach avoids any confusion like that; (as I believe I have said before: a picture is worth a thousand words! ?)

Hope that helps. ?

 

[The Highlander]

I know enough Greek to check this, and the latter translation is indeed better. There are two different words, one of which literally means oblong ("different-sided"), and the other literally means rectangular ("right-angled"). So the first translator made the same mistake you refer to. (But ultimately it's a matter of definition; apparently the Greeks defined the rhombus exclusively, like the oblong.)

But, I trust you agree that, we (now) define these things inclusively, thence all squares are rectangles and all squares are rhombi?

 

[Dr.Peterson]

Since @Dr.Peterson has given you the answers, I don't suppose you will bother to attempt what I suggested

Actually, I didn't give complete answers; I acknowledged some parts that were correct, but left some with only questions. There were still things to think about.

But, I trust you agree that, we (now) define these things inclusively, thence all squares are rectangles and all squares are rhombi?

I think I said that more or less clearly.

X ⊂ Y means that the set X is a proper subset of the set Y.

Although it doesn't really matter in this problem, it may be worth pointing out that this usage varies a bit; sometimes the symbol just means (any) subset. I'm not sure how common that is, but when it matters (and I think of it), I either ask what definition is being used, or look for clues in what was written.

You got into 'trouble' trying to explain your answers, in words, on more than one occasion and adopting this alternative approach avoids any confusion like that

On the other hand, it's a good idea to learn to use the words correctly. The difference between "every quadrilateral is not square" and "not every quadrilateral is a square", or between "a rectangle can never be a square" and "not every rectangle is a square", or between "a rhombus is a set of quadrilateral" and "a rhombus is a kind of quadrilateral", is worth knowing, even though it's at least as much English as math.

I agree that the Venn diagram can help untangle the ideas; and showing the diagram can reveal whether you understand the words. It's a good intermediate step between the question and the answer. But it might sometimes not avoid confusion, but just make it visible!

 

[chijioke]

Some are correct others are wrong.

f) has the correct answer but for the wrong reason; your thinking is faulty! (You said, at a), that: "a square is a special type of rectangle" (true) but that means that there are rectangles that are square so it wrong to then say: "a rectangle can never be a square."

(When I was in primary school our teachers taught us to call a non-square rectangle an oblong.

I think I don't understand the word 'oblong' clearly. What then is an a blong? Is a synonym for the word, rectangle?

 

[chijioke]

Some are correct others are wrong.

f) has the correct answer but for the wrong reason; your thinking is faulty! (You said, at a), that: "a square is a special type of rectangle" (true) but that means that there are rectangles that are square so it wrong to then say: "a rectangle can never be a square."

The error is noted. I should have said rather that some rectangles are squares but not all rectangles are squares?

 

[Dr.Peterson]

I think I don't understand the word 'oblong' clearly. What then is an a blong? Is a synonym for the word, rectangle?

Have you tried searching for its meaning?

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It can be used in a more general way, but one specific meaning is a rectangle that is not a square. It is not quite a synonym.

The error is noted. I should have said rather that some rectangles are squares but not all rectangles are squares?

Yes, that is correct.

And all squares are rectangles.

Here is your attachment:

​

The non-square rectangle is an oblong; the rounded rectangle and curved-end rectangle can be called oblong in the more general sense. The square is the only one that definitely is not an oblong. A non-circular ellipse or similar shape can also be called oblong.

According to one source I found, in British usage an oblong is only a non-square rectangle, and the broader definition is American only; but I am not sure of that.

 

[The Highlander]

The non-square rectangle is an oblong; the rounded rectangle and curved-end rectangle can be called oblong in the more general sense. The square is the only one that definitely is not an oblong. A non-circular ellipse or similar shape can also be called oblong.

According to one source I found, in British usage an oblong is only a non-square rectangle, and the broader definition is American only; but I am not sure of that.

Being British, I would agree with the Chambers Dictionary definition of an "oblong¹", however, we would be happy to accept that the rounded rectangles were oblong² in (general) "shape".

1. Used as a noun.
2. Used as an adjective.

 

[The Highlander]

I think I don't understand the word 'oblong' clearly. What then is an a blong? Is a synonym for the word, rectangle?

An oblong is just a non-square rectangle. So both squares and oblongs are rectangles and are wholly enclosed by Set B in the Venn diagram above, qv.
It is, therefore, useful to refer to non-square rectangles as oblongs (rather than just calling them rectangles) so that the distinction between square and non-square rectangles may be made without the misunderstanding that 'squares are not rectangles' creeping in (because children begin to think that squares and rectangles are two different things). ?

 

[chijioke]

Have you tried searching for its meaning?

Yes. But I was not satisfied with the answer I got. so I came for clarification.

Here is your attachment:

View attachment 36082​

Interesting! How were able to make the attachment appear in the forum? I tried doing same but could not succeed. I only managed to have it appear as an attachment.

 

[Dr.Peterson]

Interesting! How were able to make the attachment appear in the forum? I tried doing same but could not succeed. I only managed to have it appear as an attachment.

I just copied the image from your pdf (using the snipping tool, though there are other ways) and pasted it into my message. (You could also use the "Attach file"s button and then select" Insert...".)

A pdf can't be shown this way; but there was no reason to make it a pdf in the first place. I was wondering why you didn't do what you usually do.

 

[The Highlander]

The error is noted. I should have said rather that some rectangles are squares but not all rectangles are squares?

Or to put it concisely: "Squares \(\displaystyle \subset\) Rectangles". ?

 

[The Highlander]

This may not, of course, be true of all schools everywhere; but the misconception is certainly common. I just attribute it to the difference between everyday usage, which tends to be exclusive ("no, I asked for the rectangle, not the square!"), and technical usage, which tends to be inclusive ("this theorem applies to all rectangles, and therefore specifically to squares").

Sorry to drag this one up again but there was a repeat of the 1% Club show on the other night and I happened to turn the TV over to that channel just at the point where a primary school teacher was addressed by the host. It brought my comment about what's taught in primary schools (being unhelpful) back to mind so I capped an excerpt from the show and uploaded it here. (Please have a look; it's only about half a minute long. It's also best viewed full screen, just scroll down to the ⛶ icon.).

It struck me at the time that although he said "quadrilateral" I suspect he was actually thinking "rectangle" but, either way, he seems to believe that a square is something entirely distinct from a quadrilateral/rectangle! ?‍

It was only after I had gone to the bother of capturing the clip and uploaded it that I thought I might look up my original post (of the question concerned).

Turns out I already added a 'comment' after the 25% question in that thread pointing out my exasperation that kidds come up from the primary schools convinced that a square is not a rectangle! ?
You can view that (highlighted) here (Right-click & choose: "Open link in new tab"). ?

 

[Dr.Peterson]

You said,

When one contestant (who got it right) was asked what his job was, he said he had been a primary school teacher and, while explaining how he got it right, commented: "Well, we do a bit of Maths. Know the difference between a quadrilateral and a square." (I strongly suspect he meant to say: “rectangle”! But, either way....)

But there is a difference between a rectangle and a square, or between a quadrilateral and a square: They have different definitions, and that difference is the key to answering the question!

So I'm not sure he was saying that a square is not a rectangle/quadrilateral; and that's not really relevant to the question anyway. Now, if the question had asked, which is the smallest rectangle, and C had been 2x3, that would have been interesting!

For my view of this sort of question, see here, if I haven't already referred to it.

 

[The Highlander]

You said,


But there is a difference between a rectangle and a square, or between a quadrilateral and a square: They have different definitions, and that difference is the key to answering the question!

So I'm not sure he was saying that a square is not a rectangle/quadrilateral; and that's not really relevant to the question anyway. Now, if the question had asked, which is the smallest rectangle, and C had been 2x3, that would have been interesting!

For my view of this sort of question, see here, if I haven't already referred to it.

Indeed. but the only point I've been trying to make here is that a square is (also) a rectangle and it is unhelpful to imbue young minds with the thought that a square is not a rectangle; ie: that they are two distinctly different things (and "never the twain shall meet"!).