SETS II

 

SETS

Hello lovely students, welcome to another episode of learning. Today we are continuing our studies on sets. We are going to look at application of sets in solving problems using a Venn diagram. We will be discussing only two set problems. If you have no previous knowledge on sets, I suggest you go back to my previous post and go through.

In that post, we treated the types of sets, the properties of sets and how sets are described.  

Now let us explain what a Venn diagram is?

A Venn diagram is a visual representation of sets and their relationships. It consists of overlapping circles (ellipses) that show intersection, union and difference of sets.

We represent the universal set in a question with a rectangle, the subsets are also represented with ellipse (circles) intersecting each other.

For Example, Lets deal with these sets below.

If A and B are subsets of U, such that U= {1,2,3,4,5,6,7,8,9,10}, A= {1,3,5,7} and B= {2,3,7,9}. We will represent sets A and B using circles as shown below.

                                                                               Fig 1

In the above diagram, we can see that the values 3 and 7 have been put in at where the two circles intersect. This is the intersection and is called A∩B. 1 and 5 are placed in the other half of A and is called A only whilst 2 and 9 are in B only. Note that A = {1,3,5,7} but A only is {1,5}. B = {2,3,7,9} and B only is {2,9}. We also saw the numbers 4, 6, 8 and 10 in the rectangle but not in any of the circles. This is called Neither A nor B (None of the two) or (A U B)¹.

 

In general, lets consider that diagram below to explain a Venn diagram.

                                                                               Fig 2

In the Venn diagram above, The circle containing a and b represents the set X, the circle containing b and c is Y, b is both X and Y and d is none of the two (neither X nor Y).

A is X only and b is Y only.

Now let us solve the question below.

Given that M = {even numbers between 10 and 20}, N = {multiples of 4 less than 20} are subsets of U = {Even numbers up to 20}. Represent this information on a Venn diagram and answer the questions that follow.

a.       Find

i.                    M ∩ N

ii.                  M¹

iii.                M U N.

b.      How many members are in M only?

Solution.

U = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}.

M = {12, 14, 16, 18}

N = {4, 8, 12, 16}

So M ∩ N = {12, 16}

And M¹ = {2, 4, 6, 8, 10, 20}

M U N = {4, 8, 12, 14, 16, 18}

The diagram can be represented as shown below.

  Fig 3

APPLICATIONS OF TWO SET PROBLEMS.

Consider the question below.

1.      There are 50 boys in a sporting club. 30 of them play hockey and 25 play volleyball. Each player plays at lease on of the two games.

i.                    Illustrate the information on a Venn diagram

ii.                  How many boys play volleyball only?

iii.                How many boys play only one game?

Solution

In this section, we will introduce number of items. Instead of giving the members of set U (universal set), we will give number of members in the universal set as n(U) and so on.

We start by defining the letters we are going to use. Let us use the first letter of each game for easy identification.

So let U = boys in the sporting club

N(U) = 50.

H = boys who play hockey

N(H) = 30

And V = boys who play Volleyball.

N(V) = 25.

Let x represent boys who play both Hockey and Volleyball.

NB: Since they said each player plays at least one of the games, this means there is no player at Neither Hockey nor Volleyball column. (Part d in the fig 2)

i.                     The Venn diagram is drawn as below.

Fig 4

Now note that a =  Hockey only and it can be solved as 30 – x and b = Volleyball only which is 25 – x.

We can solve for x by adding all the three parts of the circles and equate it to 50.

i.e. a + x + b = 50.

Now substitute a with 30 – x and b with 25 – x, we have

30 – x + x + 25 – x = 50.

Group like terms and simplify

We have 30 + 25 – x = 50.

55 – x = 50

55 – 50 = x

So x is 5.

This means 5 players played both games.

That means 25 played hockey only (30 – 5) and 20 played Volleyball only (25 – 5)

ii.                  Volley ball only (a) = 30 – x. but x = 5

So a = 30 – 5 = 25

This means 25 boys played Hockey only

iii.                For only one game, we add Hockey only and Volley ball only. This has been solved above.

i.e. (30 – 5) + (25 – 5)

= 25 + 20. = 45.

So 45 players played only one game.

Now do these examples and send your answers to stakhanovite91@gmail.com or WhatsApp number 0241017418. You can also create an account here and add them as comments.

Questions

1.      P and Q are subsets of the universal set U such that

U = {counting numbers up to 18}, P = {even numbers} and Q = {Multiples of 3}.

i.                    List the members of the sets U, P and Q.

ii.                  Find the members of

a.       P ∩ Q        b. P U Q          c. P¹ ∩ Q.

2.      In a class of 40 students, 19 offer French and 25 offer Ga. 6 students do not offer any of the two subject. Illustrate this on a Venn diagram and find the number of students that offer only French.

3.      Consider these two sets.

Set A = people who own smartphones

Set B = people who own laptops.

Using a Venn diagram, shade the region representing people who own both a smartphone and a laptop? What would the shaded region represent?

a.       A ∩ B                   b. A U B         c. A – B           d. B – A.

That’s all for today. I hope you enjoyed the lesson. Let us try and find more questions on this topic that contain various information to help us master sets very well. Its your teacher DNA signing out.