Hi Folks,
I am thoroughly confused on how to approach Set theory problems involving 3 sets - Overlapping and non-overlapping.
On the BTG forum, I found many variations of the set formula.
Can u please help provide a summary of different approaches on the forum, everyone will be very much helped. Right now I see following variations of problems.
1. Sets A, B, C are all in percentages
2. Sets A, B, C are in terms of actual numbers
Questions asked are
1. Find the missing value/% of a set or combo set
eg. how many students take both Physics and Biology. ..
2. Find "exactly 2"
eg. Find how may students take exactly 2 courses.
3. Anything else ?? Pls fill in.
I would greatly appreciate if some one can provide a comprehensive summary on how to handle the different types.
Thanks a lot
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Set theory .. Request a summary of different approaches.
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vittalgmat
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I don't know that you need any formulas for these types of problems; at least, I never use any. Venn diagrams are very useful, and if you know how to fill in each section of a Venn diagram, you should be fine. A few principles:
-if the question gives percentages and no numbers, you can almost always assume you have 100 things and work with numbers;
-when filling in a Venn diagram, start from the middle (where both or all three circles overlap) and work your way outwards;
-if you do this, and at any point you don't know how many things should be in one of the sections of the diagram, then you should introduce an unknown (x);
-of course, you want to use the relationships you're given, so you can minimize the number of unknowns you use. We often know that "80 people speak English", and want to find how many people speak both English and French. Obviously you'd need more information than this to solve the problem, but if x people speak *both*, then 80-x people speak only English; there's no need to introduce two unknowns here.
-if the question gives percentages and no numbers, you can almost always assume you have 100 things and work with numbers;
-when filling in a Venn diagram, start from the middle (where both or all three circles overlap) and work your way outwards;
-if you do this, and at any point you don't know how many things should be in one of the sections of the diagram, then you should introduce an unknown (x);
-of course, you want to use the relationships you're given, so you can minimize the number of unknowns you use. We often know that "80 people speak English", and want to find how many people speak both English and French. Obviously you'd need more information than this to solve the problem, but if x people speak *both*, then 80-x people speak only English; there's no need to introduce two unknowns here.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com
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vittalgmat
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Thanks Ian ,
I use Venn Diagram, which was how I learnt in high school.
But I saw many formulas and variations.. Hence I was confused.
rgds
-V
I use Venn Diagram, which was how I learnt in high school.
But I saw many formulas and variations.. Hence I was confused.
rgds
-V
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lunarpower
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yeah, you shouldn't need a formula. if anything, you should be able to immediately IDENTIFY THE PARTS OF THE VENN DIAGRAM that correspond to each of these possibilities.
check out this diagram for reference:
https://math.sduhsd.net:8080/webMathemat ... m3Sets.jsp
(1)
this seems to refer to literally any possible question that could be asked about this diagram, so i'm not sure how to answer it. there's no advice that's sufficiently general as to apply to all possible combinations of regions!
(2)
the three chevron-shaped regions, in the intermediate color of gray (i.e., darker than the center region, but lighter than the outer regions), are the three regions corresponding to your criterion of "exactly 2".
there are a few different combinations you could use to get the values of those regions alone. i don't know that memorization of those combinations is the best route; instead, you should try to visualize the different combinations you'll see, with particular attention paid to double- and triple-counted regions.
for instance, if you add together "A and B", "A and C", and "B and C", each of which is one of the football-shaped regions encompassing one dark- and one medium-colored part, you'll get the "exactly 2" regions, plus 3 times the center region. if you then subtract out 3 times the center region, you'll be left with the "exactly 2" regions.
check out this diagram for reference:
https://math.sduhsd.net:8080/webMathemat ... m3Sets.jsp
(1)
this seems to refer to literally any possible question that could be asked about this diagram, so i'm not sure how to answer it. there's no advice that's sufficiently general as to apply to all possible combinations of regions!
(2)
the three chevron-shaped regions, in the intermediate color of gray (i.e., darker than the center region, but lighter than the outer regions), are the three regions corresponding to your criterion of "exactly 2".
there are a few different combinations you could use to get the values of those regions alone. i don't know that memorization of those combinations is the best route; instead, you should try to visualize the different combinations you'll see, with particular attention paid to double- and triple-counted regions.
for instance, if you add together "A and B", "A and C", and "B and C", each of which is one of the football-shaped regions encompassing one dark- and one medium-colored part, you'll get the "exactly 2" regions, plus 3 times the center region. if you then subtract out 3 times the center region, you'll be left with the "exactly 2" regions.
Ron has been teaching various standardized tests for 20 years.
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I guess the formula vittalgmat is talking about is:
Total = A + B + C + NONE - AB - BC - AC - 2ABC
It's basically the same thing as venn diagram i guess but imo it's faster to understand/solve questions (i am used to it if i were used to venn diagram's i guess they would be faster, no?).
Take a look at this question for instance:
https://www.beatthegmat.com/what-percent ... 26956.html
When I use the formula I did it in about 1 minute and with venn diagram in about 1 minute and 30 seconds and with the venn diagram I almost missed the last part of the formula 2ABC. I know that it's because I am inexperienced with venn diagrams but could you please comment a case where for instance venn diagrams will safe time or they would be the only way to solve the question.
Also, the following question for instance would be NIGHTMARE for me useing venn diagrams, while following the formula makes it a child's play. Again I know this is probably because I am not used to venn but isn't it just better when put down in a formula?
"Of the 200 members of certain association, each member who speaks german also speaks english. and 70 members speak only spanish. no member speaks all 3 languages. how many of the members speaks 2 of 3 languages ?
1. 60 members speaks only english
2. 20 members do not speak any of the three languages"
In other words should I learn to use venn diagrams better (a lot better i guess) or just stick with the formula that I am already used to?
Ron, Ian?
Thanks!
Total = A + B + C + NONE - AB - BC - AC - 2ABC
It's basically the same thing as venn diagram i guess but imo it's faster to understand/solve questions (i am used to it if i were used to venn diagram's i guess they would be faster, no?).
Take a look at this question for instance:
https://www.beatthegmat.com/what-percent ... 26956.html
When I use the formula I did it in about 1 minute and with venn diagram in about 1 minute and 30 seconds and with the venn diagram I almost missed the last part of the formula 2ABC. I know that it's because I am inexperienced with venn diagrams but could you please comment a case where for instance venn diagrams will safe time or they would be the only way to solve the question.
Also, the following question for instance would be NIGHTMARE for me useing venn diagrams, while following the formula makes it a child's play. Again I know this is probably because I am not used to venn but isn't it just better when put down in a formula?
"Of the 200 members of certain association, each member who speaks german also speaks english. and 70 members speak only spanish. no member speaks all 3 languages. how many of the members speaks 2 of 3 languages ?
1. 60 members speaks only english
2. 20 members do not speak any of the three languages"
In other words should I learn to use venn diagrams better (a lot better i guess) or just stick with the formula that I am already used to?
Ron, Ian?
Thanks!
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lunarpower
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well, both, of course. if you know that formula, then that's great! the point is just that you shouldn't ever need the formula, strictly speaking, although it's certainly a useful opener in some situations.Zipper wrote:In other words should I learn to use venn diagrams better (a lot better i guess) or just stick with the formula that I am already used to?
Ron, Ian?
Thanks!
don't think that you can always solve everything with that same formula, though; there are some problems that it won't crack. for instance, if i give you ab, ac, and bc, as well as abc, and ask you to find the total # that are in exactly two sets (as quoted above), your formula won't work - there are waaaaaayyy too many unknowns that you don't really need.
in this case, though, looking at the diagram and visualizing the answer will lead to the visual equivalent of (2 exactly) = ab + ac + bc - 3(abc). there's no way to get that out of the formula you have.
in fact, your formula won't work unless you are given precisely the things that appear in it, while the venn diagram is more flexible. for instance, if i tell you that X number of things appear ONLY in set b, then there's no way at all to insert that into your formula, but it's trivial to put it into the venn diagram.
--
in sum:
multiple approaches to a problem are always better than single approaches. if your question is along the lines of "should i remember approach x, or should i remember approach y?" -- then the answer is almost always "both".
Ron has been teaching various standardized tests for 20 years.
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Pueden hacerle preguntas a Ron en castellano
Potete chiedere domande a Ron in italiano
On peut poser des questions à Ron en français
Voit esittää kysymyksiä Ron:lle myös suomeksi
--
Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.
Yves Saint-Laurent
--
Learn more about ron
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Pueden hacerle preguntas a Ron en castellano
Potete chiedere domande a Ron in italiano
On peut poser des questions à Ron en français
Voit esittää kysymyksiä Ron:lle myös suomeksi
--
Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.
Yves Saint-Laurent
--
Learn more about ron
Hi Lunarpowerlunarpower wrote:yeah, you shouldn't need a formula. if anything, you should be able to immediately IDENTIFY THE PARTS OF THE VENN DIAGRAM that correspond to each of these possibilities.
check out this diagram for reference:
https://math.sduhsd.net:8080/webMathemat ... m3Sets.jsp
(1)
this seems to refer to literally any possible question that could be asked about this diagram, so i'm not sure how to answer it. there's no advice that's sufficiently general as to apply to all possible combinations of regions!
(2)
the three chevron-shaped regions, in the intermediate color of gray (i.e., darker than the center region, but lighter than the outer regions), are the three regions corresponding to your criterion of "exactly 2".
there are a few different combinations you could use to get the values of those regions alone. i don't know that memorization of those combinations is the best route; instead, you should try to visualize the different combinations you'll see, with particular attention paid to double- and triple-counted regions.
for instance, if you add together "A and B", "A and C", and "B and C", each of which is one of the football-shaped regions encompassing one dark- and one medium-colored part, you'll get the "exactly 2" regions, plus 3 times the center region. if you then subtract out 3 times the center region, you'll be left with the "exactly 2" regions.
Please the link you provided for the reference does not work.
If you can kindly check it to make sure its working i will be very grateful.
I need to understand set theory thoroughly so please help
thanks
