Unit 3: Sets

Topic 1: Venn Diagrams and Sets

Competencies

· Describe the terminology used with sets and Venn diagrams.

· Determine the placement of an element in a Venn diagram.

· Apply the notion of Venn diagrams to solve problems related to sets.

Suggested ways of teaching this topic: teacher explanation, guided practice and individual practice

Starter Activities

The teacher may lead the students in a short discussion about Venn Diagrams. This topic is where we introduce the ideas of sets and Venn diagrams. A set is a list of objects in no particular order; they could be numbers, letters or even words. A Venn diagram is a way of representing sets visually.

Teacher explanation and input

To explain, the teacher could start with an example where we use whole numbers from 1 to 10.
We will define two sets taken from this group of numbers:

Set A = the odd numbers in the group = { 1 , 3 , 5 , 7 , 9 }
Set B = the numbers which are 6 or more in the group = { 6 , 7 , 8 , 9 , 10 }

Some numbers from our original group appear in both of these sets. Some only appear in one of the sets.
Some of the original numbers don't appear in either of the two sets. We can represent these facts using a Venn diagram.

The two large circles represent the two sets.

The numbers which appear in both sets are 7 and 9. These will go in the central section, because this is part of both circles.

The numbers 1, 3 and 5 still need to be put in Set A, but not in Set B, so these go in the left section of the diagram.

Similarly, the numbers 6, 8 and 10 are in Set B, but not in Set A, so these will go in the right section of the diagram.

The numbers 2 and 4 are not in either set, so will go outside the two circles.

The final Venn diagram looks like this:

We can see that all ten original numbers appear in the diagram.

The numbers in the left circle are Set A
{ 1 , 3 , 5 , 7 , 9 }

The numbers in the right circle are Set B
{ 6 , 7 , 8 , 9 , 10 }

Guided Practice

After students understood the basic ideas, the rest of the lesson will be student’s practice filling in Venn diagrams and using them.

Lesson Notes

The intersection of sets A and B is those elements which are in set Aandset B. A diagram showing the intersection of A and B is on the left.

The union of sets A and B is those elements which are in set Aorset Borboth. A diagram showing the union of A and B is on the right.

Setting: Ten Best Friends

The teacher could take a set made up of ten best friends listing them like:

{Alemu, Balcha, Kassa, Demis, Eshet, Fire, Genet, Hadush, Imam, Jasmin}

Each friend is an "element" (or "member") of the set

Now let's say that alemu, kassa, demis and hadush play Soccerand kassa, demis and jasmin play Tennis. That is,

"Soccer" = {Alemu, Kassa, Demis, Hadush}

Tennis = {Kassa, Demis, Jasmin}

You could put their names in two separate circles:

Union

You can now list your friends that play Soccer OR Tennis. Not everyone is in that set. Only your friends that play Soccer or Tennis.

This is called a "Union" of sets and has the special symbol ∪:

Soccer ∪ Tennis = {Alemu, Kassa, Demis, Hadush, Jasmin}

We can also put it in a "Venn Diagram":

A Venn diagram is better because it shows lots of information easily.

Do you see that Alemu, Kassa, Demis and Hadush are in the "Soccer" set?
And that Kassa, Demis and Jasmin are in the "Tennis" set?
And here is the clever thing: Kassa and Demis are in BOTH sets!

Intersection

"Intersection" is those who are in BOTH sets.

In this we mean those who they play both Soccer AND Tennis ... which is Kassa and Demis.

The special symbol for Intersection is an upside down "U" like this: ∩

And this is how we write it down:

Soccer ∩ Tennis = {Kassa, Demis}

In a Venn diagram, it looks as follows:

Difference

You can also "subtract" one set from another.

For example, taking Soccer and subtracting Tennis means people that play Soccer but NOT Tennis ... which is Alemu and Hadush.

And this is how we write it down:

Soccer - Tennis = {Alemu, Hadush}

In a Venn diagram, difference of two sets can be seen as follows:

Summary So Far

∪ is Union: is in either set
∩ is Intersection: must be in both sets
- or \ is Difference: in one set but not the other

Three Sets

You can also use Venn Diagrams for 3 sets.

Let us say the third set is "Volleyball", which Demis, Genet and Jasmin play:

Volleyball = {Demis, Genet, Jasmin}

But let's be more "mathematical" and use a Capital Letter for each set:

S means the set of Soccer players
T means the set of Tennis players
V means the set of Volleyball players

The Venn diagram Union of 3 Sets: S ∪ T ∪ V is now like this:

You can see (for example) that:

Demis plays Soccer, Tennis and Volleyball
Jasmin plays Tennis and Volleyball
Alemu and Hadush play Soccer, but don't play Tennis or Volleyball
no-one plays only Tennis

We can now have some discussions on Venn diagrams of Unions and Intersections.

This is just the set S

S = {Alemu, Kassa, Demis, Hadush}

This is the Union of Sets T and V

T ∪ V = {Kassa, Demis, Jasmin, Genet}

The shaded part shows the Intersection of Sets S and V. (only Demis is available)

Thus, S ∩ V = {Demis}

The teacher may ask students as: “how about performing the following?”

take the previous set S ∩ V
then subtract T:

Expected Answer

This is the Intersection of Sets S and V minus Set T

(S ∩ V) - T = {}

Look, there is nothing there!

That is just the "Empty Set". It is a null set, so we use the curly brackets with nothing inside: {}

Universal Set

The Universal Set is the set that contains Everything that we are interested in the discussion.

Sadly, the symbol is the letter "U" ... which is easy to confuse with the ∪ for Union. Students must be reminded to be careful not to mix up the symbols.

In our case the Universal Set is our Ten Best Friends.

U = {Alemu, Balcha, Kassa, Demis, Eshet, Fire, Genet, Hadush, Imam, Jasmin}

We can show the Universal Set in a Venn diagram by putting a box around the whole thing:

Now you can see ALL your ten best friends, neatly sorted into what sport they play (or not!).

And then we can do interesting things like take the whole set and subtract the ones who play Soccer:

We write it this way:

U –S = {Balcha, Eshet, Fire, Genet, Imam, Jasmin}

U – S means: "The Universal Set minus the Soccer Set is the Set {Balcha, Eshet, Fire, Genet, Imam, Jasmin}". In other words, U – S is: "everyone who does not play Soccer".

Independent practice

o Allow the students to work by themselves and to complete a worksheet, the teachershould prepare and provide one. Monitor them for questions and to be sure that the students are working

o Students may need help with some of the later questions. Not all of the Venn diagram questions are math related. Some relate to science, and some to common knowledge, in order to allow students to practice Venn diagrams more fully. Thus, teacher shall help the class to talk about what an unknown word could be--chances are good that if one student does not know what a word means, someone else in the class will.

Example 1:Given the following Venn diagram, answer each of the following questions

(a) Which numbers are in the union of A and B?
(b) Which numbers are in the intersection of A and B?

Solution:

(a) A B = {1,3,5,7,9,6,8,10}