Upload
milo-collins
View
225
Download
0
Tags:
Embed Size (px)
Citation preview
Temperature Readings
The equation to convert the temperature from degrees Fahrenheit to degrees Celsius is:
c(x) = (x - 32) The equation to convert the temperature from
degrees Celsius to degrees Fahrenheit is:
f(x) = x + 32 Is there a temperature that has the same
reading in both Fahrenheit and Celsius? €
9
5€
5
9
FunctionSet of ordered pairs {(x,y)| xX, yY}, where every element of X is associated with a unique element of Y.
X is the domain (set of inputs) of the function.
Y is the range of the function.
The image is the set of outputs.
Some Functions to Remember
Equal Functions: f(x) = g(x)
Identity Function: f(x) = x, idR(7) = 7
Constant Function: f(x) = 3, k(x) = y0
Absolute Value Function: y = |x|
Examples of Function Rules
f(x) = -2x + 1 f(x) = x f(x) = 8 f(x) = x2 + 7 f(x) = 2x
Associate each integer with a number that is twice the integer.
Composite Functions
The composition of g with f is the function
g o f = g o f(x) = g(f(x))
Notice that g o f is obtained by first doing f and then doing g.
Properties of Some Functions
One-to-one A function is one-to-one if it never sends two
elements of the domain to the same element of the range.
Onto A function is onto if no element of the range
goes unused.
One-to-One or Onto?
Temperature functions:c(x) = (x - 32)
f(x) = x + 32
f(x) = |x|, Domain = R, Range = R f(x) = x2, Domain = R, Range = R A function that assigns each word in the English to
the first letter in the word. A function that assigns each real number with a point
on the number line. y = 2x, Domain = Z, Range = Z
€
9
5€
5
9
Inverses in Mathematics
Inverse Property Additive Inverse, Multiplicative Inverse (reciprocal) Inverse Operation Inverse Function
If a function has an inverse function, then it is 1-1. If a function is 1-1, then it has an inverse function. g -1(g(x)) = g (g-1(x)) = x, or g -1o g = g o g -1 = id(x)
Algebra Structures
Set Operation(s) with elements in the set Properties that are true but accepted without
proof (axioms) Definitions Theorems which can be proved using
axioms, definitions and other theorems
Field Axioms
Associative (+, )Identity (+, )Inverse (+, )Closure (+, )Commutative (+, )Distributive ( over +)
Binary Operation(s) on Set S
A binary operation is a function where every combination of two elements of set S results in a unique answer in the set.
M: S S S For example, addition, subtraction and
multiplication with Integers are all binary operations.
Sets and Operations
Modular Arithmetic: addition, multiplicationSet Theory Operations: , , –, Matrices: addition, multiplicationFunctions: composition as an operationSymmetries of a Triangle, RectangleComplex Numbers (a + bi): addition,
multiplication
The Game of 50
Play with the set of numbers { 1, 2, 3, 4, 5, 6 }. Player 1 chooses a number from the set. Player 2 chooses a number from the set and
writes the sum of the two numbers. The players continue choosing numbers and
writing sums. The first person to choose a number that
results in a sum of 50 wins the game.
Properties for mod(n) Activity 4.22 Activity 4.24 Activity 4.25 Activity 4.26 Which of these properties exist for mod(n),
using the binary operations + and ?Commutative,
Associative,
Identity, (If so, what is the Identity Element?)
Inverse
Matrices, M2(Z)
Matrix Addition
Matrix Multiplication€
a b
c d
⎡
⎣ ⎢
⎤
⎦ ⎥ + =
€
a+ e b + f
c+ g d + h
⎡
⎣ ⎢
⎤
⎦ ⎥
€
a b
c d
⎡
⎣ ⎢
⎤
⎦ ⎥ =
€
a• e+ b • g a • f + b • h
c • e+ d • g c • f + d • h
⎡
⎣ ⎢
⎤
⎦ ⎥
€
e f
g h
⎡
⎣ ⎢
⎤
⎦ ⎥
€
e f
g h
⎡
⎣ ⎢
⎤
⎦ ⎥
€
1,1 1,2
2,1 2,2
⎡
⎣ ⎢
⎤
⎦ ⎥
Matrix Operations
Activity 4.40 - 4.43 (addition) Activity 4.44 - 4.47, 4.48 (multiplication) Which of these properties exist for M2(Z),
using the binary operations + and ?Commutative,Associative, Identity, (If so, what is the Identity Element?)Inverse
Algebraic Structures
Set, Operation(s), Properties Group:
A group is a set G together with a binary operation * which satisfy the following:
(a) The operation * is associative for all elements of G.
(b) G contains a unique identity element, e. If x is any element of g, e * x = x and x * e = x.
(c) Each element of G has an inverse in G. If x is any element of g, x-1 is the inverse of x.
x * x-1 = e and x-1 * x = e
More Algebraic Structures An Abelian Group is a group (G, *) for which the operation is
commutative.
A Ring is a set R with two operations we will call addition and multiplication, R(+,).
A ring has the following properties.Associative, Commutative, Identity, Inverse for +
(Abelian Group for +)Associative for Distributive of over +
Examples of Rings: (Z,+, ), (Q,+, ), (R,+, ), (Zn,+, ), (M2(Z),+, )