SETS A = {1, 3, 2, 5} n(A) = | A | = 4 Sets use “curly” brackets The number of elements in Set A is 4 Sets are denoted by Capital letters 3 is an element.

Slides:

Advertisements

Similar presentations

Equations in Quadratic Form

Advertisements

< < < > > >         . There are two kinds of notation for graphs of inequalities: open circle or filled in circle notation and interval notation.

Operations on Functions

Solving Quadratic Equations.

COUNTING TECHNIQUES PERMUTATIONS AND COMBINATIONS.

Parallel and Perpendicular Lines. Gradient-Intercept Form Useful for graphing since m is the gradient and b is the y- intercept Point-Gradient Form Use.

LINES. gradient The gradient or gradient of a line is a number that tells us how “steep” the line is and which direction it goes. If you move along the.

TRIGONOMETRY FUNCTIONS

If a > 0 the parabola opens up and the larger the a value the “narrower” the graph and the smaller the a value the “wider” the graph. If a < 0 the parabola.

PAR TIAL FRAC TION + DECOMPOSITION. Let’s add the two fractions below. We need a common denominator: In this section we are going to learn how to take.

Let's find the distance between two points. So the distance from (-6,4) to (1,4) is 7. If the.

REAL NUMBERS. {1, 2, 3, 4,... } If you were asked to count, the numbers you’d say are called counting numbers. These numbers can be expressed using set.

COUNTING TECHNIQUES PERMUTATIONS AND COMBINATIONS.

DOUBLE-ANGLE AND HALF-ANGLE FORMULAS

(r,  ). You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate.

SPECIAL USING TRIANGLES Computing the Values of Trig Functions of Acute Angles.

SOLVING LINEAR EQUATIONS. If we have a linear equation we can “manipulate” it to get it in this form. We just need to make sure that whatever we do preserves.

TRIGONOMETRIC IDENTITIES

You walk directly east from your house one block. How far from your house are you? 1 block You walk directly west from your house one block. How far from.

Logarithmic Functions. y = log a x if and only if x = a y The logarithmic function to the base a, where a > 0 and a  1 is defined: exponential form logarithmic.

INVERSE FUNCTIONS.

The definition of the product of two vectors is: 1 This is called the dot product. Notice the answer is just a number NOT a vector.

Dividing Polynomials.

exponential functions

GEOMETRIC SEQUENCES These are sequences where the ratio of successive terms of a sequence is always the same number. This number is called the common ratio.

VECTORS. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude.

The standard form of the equation of a circle with its center at the origin is Notice that both the x and y terms are squared. Linear equations don’t.

A binomial is a polynomial with two terms such as x + a. Often we need to raise a binomial to a power. In this section we'll explore a way to do just.

ARITHMETIC SEQUENCES These are sequences where the difference between successive terms of a sequence is always the same number. This number is called the.

LINEAR Linear programming techniques are used to solve a wide variety of problems, such as optimising airline scheduling and establishing telephone lines.

Properties of Logarithms

Logarithmic and Exponential Equations. Steps for Solving a Logarithmic Equation If the log is in more than one term, use log properties to condense Re-write.

The Complex Plane; DeMoivre's Theorem. Real Axis Imaginary Axis Remember a complex number has a real part and an imaginary part. These are used to plot.

Library of Functions You should be familiar with the shapes of these basic functions. We'll learn them in this section.

SEQUENCES A sequence is a function whose domain in the set of positive integers. So if I gave you a function but limited the domain to the set of positive.

11.3 Powers of Complex Numbers, DeMoivre's Theorem Objective To use De Moivre’s theorem to find powers of complex numbers.

COMPLEX Z R O S. Complex zeros or roots of a polynomial could result from one of two types of factors: Type 1 Type 2 Notice that with either type, the.

Sum and Difference Formulas. Often you will have the cosine of the sum or difference of two angles. We are going to use formulas for this to express in.

Solving Quadratics and Exact Values. Solving Quadratic Equations by Factoring Let's solve the equation First you need to get it in what we call "quadratic.

Surd or Radical Equations. To solve an equation with a surd First isolate the surd This means to get any terms not under the square root on the other.

COMPOSITION OF FUNCTIONS “SUBSTITUTING ONE FUNCTION INTO ANOTHER”

VECTORS. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude.

Remainder and Factor Theorems. REMAINDER THEOREM Let f be a polynomial function. If f (x) is divided by x – c, then the remainder is f (c). Let’s look.

Dividing Polynomials Using Synthetic Division. List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put.

INTRODUCING PROBABILITY. This is denoted with an S and is a set whose elements are all the possibilities that can occur A probability model has two components:

The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms &

Let's just run through the basics. x axis y axis origin Quadrant I where both x and y are positive Quadrant II where x is negative and y is positive Quadrant.

We’ve already graphed equations. We can graph functions in the same way. The thing to remember is that on the graph the f(x) or function value is the.

The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms &

(r,  ). You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate.

Sets Finite 7-1.

SIMPLE AND COMPOUND INTEREST

Systems of Inequalities.

RATIONAL FUNCTIONS II GRAPHING RATIONAL FUNCTIONS.

THE DOT PRODUCT.

Matrix Algebra.

Absolute Value.

INVERSE FUNCTIONS.

SETS Sets are denoted by Capital letters Sets use “curly” brackets

SIMPLE AND COMPOUND INTEREST

INVERSE FUNCTIONS Chapter 1.5 page 120.

Solving Quadratic Equations.

INVERSE FUNCTIONS.

Symmetric about the y axis

exponential functions

Symmetric about the y axis

Presentation transcript:

SETS A = {1, 3, 2, 5} n(A) = | A | = 4 Sets use “curly” brackets The number of elements in Set A is 4 Sets are denoted by Capital letters 3 is an element of A 7 is not an element of A

A set is a distinct collection of objects. The objects are called elements. {1, 2, 3, 4} = {2, 3, 1, 4} Order does not matter. If a set contains the same elements as another set, the sets are equal. {1, 3, 2, 3, 5, 2} We never repeat elements in a set. {1, 3, 2, 5} This symbol means "is a subset of" This is read "A is a subset of B". A  B A = {1, 2, 3} B = {1, 2, 3, 4, 5} {1, 2, 3, 5} In ascending order

If a set doesn't contain any elements it is called the empty set or the null set. It is denoted by  or { }. NOT {  }  It is agreed that the empty set is a subset of all other sets so: List all of the subsets of {1, 2, 3}.  Notice the empty set is NOT in set brackets. {1}{2}{3}{1, 2}{1, 3}{2, 3}{1, 2, 3}

? ?  Number of Elements in Set Possible SubsetsTotal Number of Possible Subsets 1. {A} {A}   2 2  2. {A, B} {A, B} {A} {B}   4 4  3. {A, B, C} {A, B, C} {A, B} {A, C} {B, C} {A} {B} {C}  8 8  4. {A, B, C, D}{A, B, C, D} {A, B, C} {A, B, D} {A, C, D} {B, C, D} {A, B} {A, C} {A, D} {A, B} …… {D}  16 The number of possible subsets of a set of size n is ? 2n2n

A  BA  B This is the union symbol. It means the set that consists of all elements of set A and all elements of set B. = {1, 2, 3, 4, 5, 7, 9} Remember we do not list elements more than once. A  BA  B This is the intersect symbol. It means the set containing all elements that are in both A and B. = {1, 3, 5} A = {1, 2, 3, 4, 5} B = {1, 3, 5, 7, 9}

These sets can be visualized with circles in what is called a Venn Diagram. A  BA  B AB Everything that is in A or B. AB A  BA  B Everything that is in A AND B. AB

Often will have a set that contains all elements that we wish to consider. This is called the universal set. All other sets are subsets of this set. Universal Set A B A  B =  There are no elements in both A and B. When this is the case they are called disjoint sets. This means the complement of A, and means the set of all elements in the universal set that are not in A. A A

100 people were surveyed. 52 people in a survey owned a cat. 36 people owned a dog. 24 did not own a dog or cat. Draw a Venn diagram. universal set is 100 people surveyed CD Set C is the cat owners and Set D is the dog owners. The sets are NOT disjoint. Some people could own both a dog and a cat. 24 Since 24 did not own a dog or cat, there must be 76 that do. n(C  D) = 76 This n means the number of elements in the set = 88 so there must be = 12 people that own both a dog and a cat Counting Formula: n(A  B) = n(A) + n(B) - n(A  B)

Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar