How To Find Maximum And Minimum Values In Sets

Set Theory- Maximum and Minimum Values

Sunday, Oct 11th, 2022

Set Theory Concepts

The concepts of Set Theory are applicable not only in Quant / DI / LR but they can exist used to solve syllogism questions as well. Let united states of america outset sympathize the basics of the Venn Diagram earlier we movement on to the concept of maximum and minimum. A large number of students become confused in this so I have listed out each area separately.

A venn diagram is used to visually represent the human relationship between various sets.

What do each of the areas in the figure represent?

I – but A;

II – A and B but non C;

III – Simply B;

4 – A and C only not B;

V – A and B and C;

VI – B and C just not A;

7 – Only C

n(A U B U C) = n(A) + n(B) + n(C) — n(A due north B) — due north(A n C) – due north(B n C) + northward(A due north B n C)

Equally per the higher up diagram,

n(A) = I + II + IV + V

n(B) = II + Three + V + VI

due north(C) = IV + Five + VI + VII

due north(A due north B) = Two + V

n(B n C) = 5 + Six

n(C n A) = Four + V

north(A due north B n C) = V

n(A U B U C) = I + II + III + IV + V + Six + Seven

Note: While doing such questions, it is advisable that yous accept the to the lowest degree no. of variables to make full up the empty space. Every bit a exercise if north(A n B due north C) is missing, take that as 'x' and continue.

For Maximum and Minimum of values, the key bespeak to annotation is:

If you allot a value to the intersection, information technology will go added to all the private sets only will bring down the full.

Example: In a survey it was found that 80% like tea whereas 70% like coffee. What is the maximum and minimum number of those who similar both?

Ans: First thing to note is that no information is mentioned about the people who don't like either of them. And so that value is flexible and tin can change.

n(tea) = 80

northward(coffee) = 70

north(full) = 100 {This includes those who similar neither.}

n(tea northward coffee) = ??? {Nosotros don't know this value and it is flexible}

If we want to maximize those who similar both, we have to maximize the value in the intersection. So, we have to minimize the value of the union.

northward(tea n java)_max = 70 {It is express by the college of the ii values}

In this case, our venn diagram will await something similar this (Cerise is tea & Purple is java):

In this case 20% of people like neither tea nor coffee.

If nosotros desire to minimize those who like both, we have to minimize the value in the intersection. Then, we have to maximize the value of the wedlock. We know that the maximum possible value of the union ie n(tea U coffee) = 100

And then, nosotros demand to figure out the surplus : n(tea) + n(coffee) = 80 + lxx = 150.

The surplus is = 150 – 100 = 50

So, the value of the intersection = value of the surplus = 50

This could have also been obtained past the formula

n(a U b) = n(a) + n(b) – n (a n b)

In this case, our venn diagram volition look something similar this (Red is tea & Majestic is java):

In this case, there is no one who likes neither coffee nor tea.

Let us await at a slightly more than complicated problem when nosotros accept to deal with three sets and the value of union of the sets is fixed.

Case: In a survey it was constitute that 40 % like tea, 50 % like coffee and 60 % similar milk. Every person likes at least one of the three items – tea / coffee / milk. What are the maximum and minimum possible values of those who like all three?

Solution: Currently our Venn Diagram looks like this:

Offset of all, we need to figure out the surplus.

Surplus = twoscore +50 + 60 – 100 = 50

The surplus should exist taken care of by adding to the intersection of all iii or whatever of the two.

If we want to maximize those who like all 3, we need to maximize the intersection of all three. Adding '1' to the intersection of all three takes care of a surplus of '2'. To accept intendance of a surplus of l, we need to make

n (tea n coffee due north milk) = 25

Note: If the marriage of the sets was not fixed i.due east. the line 'Every person likes at least one of the iii items – tea / java / milk' was non given in the question then the respond would have been xl.

Our venn diagram will now look similar this:

If we want to minimize those who like all three, we need to minimize the intersection of all iii. Only nosotros have to accept care of the surplus of 50. We tin can practice that adding them to the intersection of any two of them. Adding 'ane' to the intersection of two sets, takes care of a surplus of '1'. So,

n (tea n coffee n milk)_min= 0

We can have intendance of the surplus fifty in many means by adding them in whatsoever gild to the intersection of two sets. Three of those many means are given beneath in Venn Diagrams.

I hope that this will help you in solving problems related to Gear up Theory.

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