Unit 1 - Numbers

Question 1

What does notation mean?

1.1

Answer 1

Notation: a system of marks, signs, figures, or characters that is used to represent information
((Two types: Set Notation (2A) and Interval Notation (2C) ))

Question 2

What does set mean?

1.1

Answer 2

Set: a collection of numbers and objects
- Within each set, each number/object is called an element or member
- The symbol for element is ∈

Question 3

In set notation, how do we indicate if something is an “Element of” versus “Not an element of”?

1.1

Answer 3

• In order to indicate if something is an element of the set or not, we name the element, then the symbol for element of ‘E’ (or for not an element of, an ‘E’ with a line down the centre) , then the set symbol

Question 4

What is set notation?

1.1

Answer 4

• Requires a Capital Letter to represent the set
• NOTE: All elements in a set are separated by a comma, use squiggly bracket around the elements
• Defining allows you to use the set to represent those elements throughout the problem

Question 5

What are the types of Sets?

1.1

Answer 5

Finite Set
- A set of numbers in which the elements do not continue until forever (Eg amount of pens in a case)

Infinite Set
- A set of numbers in which the elements do continue until forever (Eg All odd numbers to exist)

Null/Empty Set
- Has no elements at all
(Eg Days of week with less than 5 letters)

Question 6

What are natural numbers?

1.1

Answer 6

Natural Numbers: counting numbers [positive]
(1, 2, 3, 4…)
- Symbol N with double line going diagonally down

Question 7

What are whole numbers?

1.1

Answer 7

Whole Numbers: counting numbers plus zero [positive still]
(0, 1, 2, 3, 4…)
- Symbol W with double line on either side coming diagnally down

Question 8

What are integers?

1.1

Answer 8

Integers: positive and negative whole numbers [includes zero]
(…-3, -2, -1, 0, 1, 2, 3…)
- Symbol Z with double line on middle line coming diagonally down
- Can also have Z+ or Z- for positive or negative integers

Question 9

What are rational numbers?

1.1

Answer 9

Rational Numbers: solved decimals in which it TERMINATES or REPEATS predictably
- Can be represented as fraction
- Symbol Q with double line curved on either side

Question 10

What are irrational numbers?

1.1

Answer 10

Irrational Numbers: solved decimals that DO NOT TERMINATE and are UNPREDICTABLE
- Eg √2, or π
- No symbol

Question 11

What are real numbers?

1.1

Answer 11

Real Numbers: All the numbers that possibly exist (between negative and positive infinity)
- Symbol R with double line on straight side

Question 12

Why is interval notation better than set notation?

1.2

Answer 12

• Set Notation takes too much time if there are a lot of elements in a set
• Interval Notation can be more efficient as; it describes all the values within the set easily and quickly for continuous data

Question 13

How do we express interval notation on a number line?

1.2

Answer 13

> < Less than or greater than is expressed with an open circle
≤ ≥ Less than or equal to and greater than or equal to are expressed with a closed circle (colured in)
- For continous data (real numbers) we can draw a line accross to show it includes all the number between, for discrete data we have to mark each point

Question 14

What the steps for interval notation?

1.2
(Refer to notes for how to draw)

Answer 14

Step 1: Introduce the variable
- The letter/variable represents the context of the set

Step 2: Analyze the limitations
- Does it have an upper and a lower limit?
- Does it “include” or not?

Step 3: Analyze the type of number
- Frame the limitation with the type of number
- **If there is no number type, always assume Real Numbers

Question 15

What is a subset?
What is the symbol?
1.3

Answer 15

A secondary set that contains elements of the primary set.
The symbol for ‘is a subset of’ is a squashed c on top of a horizontal line.
The symbol for ‘‘isn’t a subset of’ is the same, with a diagonal line through it.

Question 16

What are complement sets?
1.3

Answer 16

The complement of a set is the prime of it, all the elements not in the set that are in the universal set (Eg complement of A is A’)

Question 17

How can we use venn diagrams to express a set?
1.3

Answer 17

We draw a rectangle, and label it U (for the universal set) on the bottom left corner, a circle is drawn to represent a set

Multiple circles can be drawn, overlap if the elements do, to show all elements within the universal set

If there are elements in the universal set that aren’t a part of the subset, they are drawn around the circles in the rectangle

Question 18

What is an intersection?
1.4

Answer 18

Of sets A and B, an intersection consists of all elements which are in both A and B

Notation: A ∩ B

Question 19

What is a union?
1.4

Answer 19

Of sets A and B, a union consists of all elements which are in A OR B (or both)

Notation: A ∪ B

Question 20

What does disjoint mean in terms of sets?
1.4

Answer 20

Two sets are disjoint or mutually exclusive if they have no elements in common.

Notation: A ∩ B = ∅ (Translation, A intersecting B has no elements)

Question 21

What are the first 2 identities?
(involving a set and its complement)
1.4

Answer 21

1. A ∪ A’ = U
   Any set in union with it’s complement is equal to the universal set that its from.
2. A ∩ A’ = ∅
   The intersection of an element and it’s prime is a null set

Question 22

What are the second 2 set identites?
(involving 2 different sets)
1.4

Answer 22

3. (A ∪ B)’ = A’ ∩ B’
   The complement of the union of two sets is equal to the set 1 prime intersecting with set 2 prime.
4. (A ∩ B)’ = A’ ∪ B’
   The complement of the intersection of two sets is equal to the set 1 prime in union with set 2 prime.

Question 23

How do you show the number of elements in a set in a venn diagram?
1.4

Answer 23

If a number in a set is in brackets, it is the number of elements in that set.

Question 24

How can we solve problems with venn diagrams?
1.5

Answer 24

Label set 1 with the letter a, label the overlap of set 1 and 2 with b, and label set 2 with C

Use the question to answer what a+b equals (how many is it telling you are in set 1), do the same for b+c

Then do A+2b+C (adding the above together) minus a+b+c which can be found by taking the total number minus the number not in either set

Once you’ve solved the overlap, subtract it from the amount that is in set 1 and set 2 to find the amount ONLY in each of them

Question 25

What does relations mean?
What is the difference between continuous and discrete relations?
1.6

Answer 25

Relations: Any set of points that collect two variables

Continuous: Includes points in between

Discrete: Doesn’t include points in between

Question 26

What is domain?
What is range?

How do we express domain?
1.6

Answer 26

Domain: The set of possible values that x may have (x = left to right)

Range: The set of possible values that y may have (y=up to down)

For discrete functions: list them in set notation (eg. x = {5,7,8}

For continuous function: write in interval notation
(eg. {x|6≤x<9, xER}, x such that 6 is less that or equal to x less than 9, where x is an element of all real number.)

Question 27

What is a function?
1.7

Answer 27

A relationship in which no two different ordered pairs have the same first number.

*For a graph or number set, every individual x value corresponds to only one y value

Note: y values may appear for more than one x value

Question 28

What is the vertical line test?
1.7

Answer 28

To test if something is a function.

If the line cuts the graph more than once, its not a function.

Question 29

Function Notation
and important points
1.8

Answer 29

f: x → 2x -1
Function of f such that when I put x in, ____ is the output

Simplified to: f (x) = 2x - 1

Important points:

1. f(x) is read as f of ‘x’ and is the value of the function at any value of x
2. f(x) is synonymous with ‘y’ from previous equations for our purposes
3. f(x) is sometimes called the image of ‘x’

Question 30

Restricted Domains - Undefined values
1.8

Answer 30

1/x, denominator can’t be zero
√-x, can’t have root of a negative number