Sets Part1

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Contents 1 Notes 1.1 Learning Objectives (Syllabus) 1.2 Prior Knowledge 2 Symbols and notations 3 Identity & Inverse Laws 3.1 Of Apples and Oranges 4 Exercise 1 5 Commutative Laws 6 Associative Laws 6.1 Bracket Matters 7 Combining Commutative and Associative 8 Exercise 2 9 Distributive Laws 9.1 How to expand 9.2 How to factor 9.3 Using Distributive laws 10 Exercise 3 11 Associative vs Distributive vs Identity 11.1 Comparison 12 Combining (associative + commutative) with distributive 12.1 Of Apples and Oranges (Part 2) 13 Exercise 4 14 Exercise 5

Notes

• This part is going to focus solely on Set Algebra (specifically the laws of the algebra of sets). Other parts might be better covered in more appropriate sections (probability, number intervals, etc. or perhaps an introductory section to sets).
• I teach Set Algebra as the very first topic to my lower 6 students, the main reason being that it requires almost zero prior knowledge. It also has relatively few things to concentrate on and only one type of question (of proving). These factors make it the ideal gauge to the students ability to learn maths. In my opinion, any student who find this part difficult to deal with (even after trying out all the material here) should reconsider his options in taking STPM maths. Because it shows the inability to deal with laws and learning how to use them, and without that ability, there is really nothing left that can be done in paper 1.

Learning Objectives (Syllabus)

• understand the concept of set and set notation (not covered here)
• carry out operations on sets (not covered here)
• use the law of the algebra of sets

Prior Knowledge

• Almost none, except knowledge of Venn diagrams that helps in remembering the identity & inverse laws

Symbols and notations

: Subset of Used for sets, ex :

: Element of Used for elements, ex :

: Null Set/ Empty Set for our purpose here (but not the right definition at all), it might help to remember this as NOTHING

: Universal Set EVERYTHING

: Union OR - in the context of Venn diagram, means we take ALL areas that are in OR in (or both)

: Intersect AND - in the context of Venn diagram, means we take ONLY areas that are in AND ALSO IN

: Complement of OUTSIDE - in the context of Venn diagram, means we take areas OUTSIDE of

Identity & Inverse Laws

a)

b)     any set union(OR) or intersect(AND) with itself will give back itself

c)     union(OR) with NOTHING set gives us back that set

d)     intersect(AND) with NOTHING gives us NOTHING

e)     union(OR) with EVERYTHING gives us EVERYTHING

f)     intersect(AND) with EVERYTHING set gives us back that set

g)     INSIDE union(OR) with OUTSIDE gives us EVERYTHING

h)     INSIDE intersect(AND) with OUTSIDE gives us NOTHING

i)     OUTSIDE of OUTSIDE is INSIDE

j)     Complement of EVERYTHING is NOTHING

k)     Complement of NOTHING is EVERYTHING

• The extra explanation I give might or might not help understand them, so just ignore them if you already understand why each law are as such. You do have to find a way to make yourself understand why though, blindly memorizing them will bring you nowhere in the long run.
• The set can be substituted with any other set or any combination of set

Of Apples and Oranges

In any formula, it is of utmost importance to identify the variable in it. For example, in the law , the part "" is fixed whereas can be replaced, which means is the (only) variable in this formula.

You can use "apple" or "orange" to represent those variables.

On the web, I will use to represent "apple" and to represent "orange".

Since this formula has only one variable, the formula becomes .

Now, on your paper, draw the "apple" large enough for you to write inside it. Now, if we write inside the first "apple", means we must also write inside the second apple. Thus, we can conclude it is not just , but , , and so on. So it really is ANY set union with null set gives us back THAT SAME set. So you really should remember it as "a set""Null set"="That same set".

The real value of using "apple" or "orange" is that you can draw them as big as you like, and you can write whatever inside it. So you might get stumped with at first. But just draw an "apple" around the , and what do you get? which will be equals to . Which just means .

Try these examples :

Now lets look at . There is still only ONE variable.

Now try :

Hint : Draw apple around if cannot see.

Exercise 1

Simplify the following

Note : You MUST NOT refer back to the laws when doing this part. You HAVE TO train yourself. Give yourself sometime to look through the formulas first, then only start doing from beginning till end, and finally only check your answer.

1)

2)

3)

4)

5)

6)

7)

8)

9)

Commutative Laws

a)

b) CAN change position between two sets

Associative Laws

a)

b) CAN change order IF same operation

Note: a) Can also be written as and b) as but DONT do this in working (only do it if the final answer given in that form)

Bracket Matters

• Order of brackets -
• Only 2 sets in a bracket, every pair of sets MUST have bracket

Example :

• CORRECT
• CORRECT. All the brackets shown here ARE necessary and cannot be omitted. and are considered a pair, which is why it must be surrounded by another bracket before operation with any other set(s). Same thing applies for and
• WRONG! Never ever have 3 sets with different operations without a bracket It should be written as either or (which are different)
• NOT encouraged in working
• Correct or Wrong? Look carefully... It's WRONG. Look carefully and you'll see , 3 sets with different operation

Combining Commutative and Associative

• Combining both, thus we CAN change position AND order ( meaning, we can rearrange the sets as we like, and put the brackets as we like), but ONLY IF SAME operation
• NEVER do that when different operation or when have complement outside the bracket

Example

• CAN. Same operation.
• CAN. Same operation.
• CANNOT. Different operation
• CANNOT. Even if it seems to be same operation, complement outside the bracket will change the operation

Example 1:

• Lets analyze it first. What do you see? 3 different sets with same operation. Furthermore, we see and
  • So we can forget about the brackets first, and we rearrange them. But why would we even want to do that? Because we know and can be simplified, thus we bring it together.
  • So without that bracket, it would look like and we rearrange it to , lastly we put the brackets surrounding the and , since that is what we will simplify first.
• Thus, the working will look like this
• 
• 
• 

Example 2:

• 

Try it on your own first! Answer is

Working

• ... always start with the question
  • ... here we rearrange and "re-bracket" the sets. Remember we do this only because ALL operations are the SAME here.
  • 
  • 

Exercise 2

Simplify the following using set algebra

Note : Again you MUST NOT refer back to the laws or notes when doing this part. Train yourself. Give yourself sometime to look through the formulas first, then only start doing from beginning till end, and finally only check your answer (and also working if needed)

1)

• • 

2)

• • 

3)

• • 

4)

• • 

Distributive Laws

a)

b) "Expand" (not the correct term, but easier to imagine) if different operations

How to expand

Be VERY careful of the operations

• and NOT
• Compare with to easily remember it. You can even draw arrows (from to and from to ) as guide.

If needed, rearrange using commutative law BEFORE expanding

• = Correct? WRONG. It's easy to spot the mistake now, but this mistake is quite a common careless mistake, especially when doing things too quickly. Thus, spend sometime to rearrange it.
• =

How to factor

It's considerably harder to use the law from right to left, and we must firstly identify the forms which can be "factored" (not the correct term, but easier to imagine).

Lets first rewrite the laws:

a)

b)

Now, what do we see on the left hand side?

• First, there must be a set which is present in BOTH of the brackets ( in this case it is ).

• Secondly, the operations MUST be arranged in either or . Others DO NOT work.
  • or or etc. DO NOT work.
  • or  ? Can be done using commutative and associative laws.

Example

• CAN. is in both brackets and the operations are
• CANNOT. There isn't any same set on both brackets.
• CANNOT. The operations are not in the right order.

Example : Simplify the following

• Can this be "factored"? Yes. The operations are in the right order and the set is in both brackets
  • First, lets rearrange it so that the same set is in front of both brackets.
    • 
  • The next step can be tricky, in regards to the operation. We know it would look like
    • 
  • but it's easy to make a mistake with the operation. Making the mistake won't be a problem though if we know an error-proof way to check. In this case, we simply in our mind "expand" back whatever we write and check whether it is correct. We do this because it is alwasy easier to expand rather than factor.
    • Thus, if we carelessly write , we know it is wrong, cause it will give us
    • 

Now try Ans:

Using Distributive laws

Now let's put this new laws to use:

Example 1:

• Analyze Different operations. We see and . Thus, we expand.
  • 
  • =
  • = ...don't forget the earlier laws!
  • =

Example 2:

• Analyze Same set Thus, we factor.
  • 
  • = ...careful the next step!
  • =
  • =
  • =

Exercise 3

Simplify the following using set algebra

Note : Learn to analyze the question and see why they can/need to be expanded/factored

1)

• • 

2)

• • 

3)

• • 

4)

• • 

5)

• • 

6)

• • 

7)

• • 

8)

• • 

Associative vs Distributive vs Identity

Let's take a breather from learning new things and spend some time differentiating the things we have learnt.

First, lets look at the fundamental differences between the associative(combined with commutative) and distributive laws.

We use associative

• when the operations are the same
• by rearranging and re-bracketing the terms

We use distributive

• when the operations are the different
• by "expanding"

Thus is done through associative

while is done through distributive

Identifying what laws the situation allow us to use will of course then determine what we do next.

So, for example, when we see , we see different operations, thus we MUST use distributive law, and it becomes .

It is surely WRONG to write .

And if we see , we see same operations, thus we simply rearrange and re bracket it to .

It is fundamentally WRONG to make it (as if expanding), even if it leads to the same answer.

If you have trouble (either understanding or agreeing) with the few above lines, don't be too hard on yourself, take a breather, and look at the lessons again with a fresh mind. You are simply missing the whole point of it. It could be because you concentrate too much on the steps, too much of trying to memorize things, and care too much about finding the answer. That shows you have a fundamentally wrong method of learning maths. Trust me when I say that these method, even if it could help some students to score A1 in add maths, have zero value in STPM Maths. It could also be because you consciously/unconsciously "create" your own laws that seem logical and use them instead of the stated laws.

Comparison

3 sets, SAME operation DO NOT expand! We rearrange.

3 sets, DIFFERENT operation DO NOT rearrange! We "expand".

We rearrange.

We factor.

 ?? Look carefully... It's an identity.

 ?? There is no law that have the form , thus we have to look at it a different way.

Combining (associative + commutative) with distributive

Of Apples and Oranges (Part 2)

When we have a situation that don't suit the formula, we have two choices

• A) Derive a new formula
• B) Simplify the situation so that it fits the formula

Answer: MOST of the time, B) is a much wiser choice. In other words, simplify the complicated, DON'T complicate the simple. Good advice even for life in general. ;-)

So let's look at the above problem.

There are 4 sets, but the order of operation does not allow us to simplify by "factoring" (distributive). Thus, we have to simplify it by looking at it as 3 sets, rather than 4.

We do this by grouping certain sets and assume it to be a single set for further manipulation. We can again use the apple "" to help us imagine. Thus, the question becomes

OR which we can manipulate using associative and distributive laws respectively.

Before continuing, take note that at this stage, we HAVE NOT used any laws, we are just putting the symbol around the bracket which is already there. We CANNOT put the apple in the middle to get , for example.

So now lets compare with . The first only allow us to rearrange and re bracket, whereas the second needs us to expand. Of course, we will choose the first.

Let's see how we are going to simply

We see the two 's

Thus we know we can try the rearranging and rebracket method, but first, we must assume the last two sets as a single one

So in effect, what we have now is 3 different sets with the same operation

Where we have removed the front bracket (but remember the bracket inside the "apple" CANNOT be removed)

So now, what do we do? Consider the simpler case of ,

We have paired with since we know it will simplify.

So for , which two sets do we pair to rebracket?

Since there is a inside the "apple" , thus we pair with

and we continue with distributive and simplify.

Thus, the full working will be

• 
• 
• 
• 
• 
• 

Now, try simplifying

Did you get it? If not, try again on your own first!

Working

• 
  • ... this would be the most important step. Make sure you understand the reason we do it
  • 
  • 
  • 
  • 
  • 

Exercise 4

Simplify the following using set algebra

Note : Make sure you understand fully the above two examples before attempting this.

1)

• • 

2)

• • 

3)

• • 

4)

• • 

5)

• • 

6)

• • 

Exercise 5

IMPORTANT NOTE

This is a revision exercise, and EVERY question here is just a REPEAT from the earlier exercise, however, they are jumbled up. As such, this is a practice of analyzing questions and choosing what method to use. It is VERY important that you try to do this exercise WITHOUT referring to any notes/ previous exercise. Doing so defeats the very purpose of this exercise. Instead, take some good time to look through ALL material before this, and then try these questions from beginning till the end, and see whether you can get all of them. Take note also that it is not only the answer you need to check, your working must be correct also. If you make a lot of mistake, its best to review the material again BEFORE continuing to part 2.

Simplify the following using set algebra

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

12)

13)

14)

15)

16)

17)

18)