Sets and Venn Diagrams. Some word problems concern sets and members of sets.
So let's just say that the topic of sets overall is potentially complicated, because human beings can belong to several sets at once.
任何人都是某种性别,他们有一定的母语,一类社会经济阶级,他们有一个最喜欢的冰淇淋味道等。
There are hundreds of ways to classify human beings. It's a very complicated topic.
幸运的是,该测试侧重于几个相对简单的场景。第一个设置场景,最简单,是有两组的,而且
each member of the collection may belong to either group individually, or to both groups at once, or to neither.
例如,在高中,两组可能在棒球队和乐队中。
Obviously, some students will be on the baseball team, others will be on the band. There'll be a few talented students who are on both the baseball team and
in the band, and, of course, there'll always be some who are not participating in either.
This scenario is often represented by a Venn Diagram. A basic Venn Diagram consists of two overlapping circles inside a rectangle.
Occasionally, the test will give this diagram as part of the question. If not, students often find it helpful to sketch one for their calculations.
Elements inside one circle are in one of the groups, and elements in the overlap region are in both groups.
Notice that this diagram consists of four distinct regions. So the red region, A, this is inside the left circle but
not inside the right circle. So these are all the people in the green circle, inside the green circle,
but outside the purple circle. So they're in the green circle only, not in the purple circle.
B is the overlap group. They're in both circles.
C, these are all the people in the right circle, not inside the left circle, so these are in the purple circle only, and not in the green circle.
And then D are all the people outside of both circles. So notice that these four regions, when we add them up, this adds up to the total.
Either the total number of students, or the total number of people that are in this particular study.
Notice that the way information is given here, the verbal information, can be very tricky.
Let's say for the sake of argument that the left circle means in the band, and the right circle means on the baseball team.
If the question says 35 students are in the band, well that means A + B = 35. Because when we say 35 are in the band, we're counting all the band members.
一些乐队成员的乐队。另外一些er band members happen also to be on the baseball team, but
they're all in the band. And so both of them are counted together as in included
when we say 35 are in the band.
But if we change the wording slightly, we say, 35 students are in the band only. Well, this means that A = 35.
Now we're talking about the students that are not included in the purple circle, they're only in the green circle, and so that's region A alone.
So it's very important to be extremely careful when you read the wording of these questions,
because subtle changes in the wording can mean profound mathematical consequences.
A typical sets question of this variety almost always gives the size of the whole group, which would equal the sum of these four regions.
The other information will provide some other regions, and from addition or subtraction, we can determine the rest.
Here's a simple practice question. Pause the video and then we'll talk about this.
Okay, so we'll start out by drawing a Venn Diagram. And the green circle on the left, that's going to represent the band.
The purple circle on the right, that's going to represent the baseball team. We know that 60 are in the band, so that's A + B = 60.
35 are on the baseball team, That's B + C = 35. Notice that the B's are counted twice.
They're counted once for their band membership, once for their baseball membership.
25, that's the region outside. Well that's interesting.
So, if outside the circle there are 25, and there are 100 students all together, it means that all the people inside the two circles that must add up to 75.
Because 75 plus 25 equals 100, and so it means that A + B+ C = 75. Well it's interesting because B plus C by itself, we see at the top that's 35.
So we can replace B plus C with 35, and that allows us to solve for A, A = 40. Now we can plug that into the first equation.
A + B = 60, so 40 + B = 60, and B = 20, and that's our answer.
Notice, incidentally, if you're curious, A = 40, B = 20, C = 15, D = 25.
And those together add up to 100. But our answer here is B = 20.
这是一个稍微复杂的问题。暂停视频,然后我们会谈谈这个。
Okay, so, French and Spanish. Let the left circle equal French and the right circle equal Spanish.
And what I'll say here, unfortunately I need a lot of work to work out the Algebra on this problem.
So I had to go to a slide that didn't have text of the question on it, but if you look under this video you should see the text of the question printed, and so
you can refer to that as I'm talking about this if you wanna read along with me.
So, the first thing it says, is just as many as study neither as study both. Well, the both region is B, the neither region is D, so B=D.
好的,这很重要。第二个我们被告知学习西班牙语的四分之一也研究法国人。
Well those who study Spanish, that's B plus C so one quarter of B plus C Equals the ones who also study French,
so those are the people in the overlap group.
So one quarter B + C = B. I'm just gonna clear the fractions by multiplying that equation by 4, and
then it turns out I can subtract B from both sides, and I get C = 3B. And at this point, that gives me a bit of insight.
这表明现在的方法解决方案,因为我们're able to express D in terms of B.
Now we can express C in terms of B. If we could express A in terms of B, then we could solve for the value of B, and
that would allow us to solve for everything. Okay, so let's think about A.
The final sentence is that the total number who study French is 10 fewer than those who study Spanish only.
因此,学习法国人A + B的总数仅仅是西班牙语的10。西班牙语只有c,所以a + b = c-10。
所以现在我们只是将插入C = 3b,从两侧减去B,所以a = 2b-10。
So now we've expressed A, C, and D all in terms of B. This is important, because we know A + B + C + D = 200, and
we can substitute those other letters for expressions involving B.
And so we'll just add 10 to both sides so we'll get 210 on the left, and then we have 2B + B + 3B + B, that adds up to 7b.
So 7B = 210, divide by seven, we get B = 30 so now we have the value of B. Well, we're looking for how many students study French only so A,
we're looking for, we'll just plug that into the expression for A. A = 2B- 10, or 2 * 30- 10.
60- 10 = 50, and that's our answer. In summary, Venn Diagrams can be helpful in problems with two overlapping sets.
The problem may give a Venn Diagram, but if it doesn't, one if often helpful for solving this kind of question.
It's good to sketch it on the scrap paper, and remember to be careful interpreting wording.
如果它在X中说,就像乐队中的每个人,或者每个人都在学习法语,或者那样,总是包括那些也是另一组的人。
It includes all the overlap, but if it says all those in X only, all those studying French only,
all those on the baseball team only, then that excludes the overlap group. We have to be very careful interpreting the work.
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