| Section | Conponents | ||
|---|---|---|---|
| Section 1 | Introduction to Numbers and Numbers Systems | ||
| Section 2 | Binary Numbers, Hexadecimal Numbers, Octal Numbers | ||
| Section 3 | Set Theory | ||
| Section 4 | Logic & Logic Networks | ||
| Section 5 | Functions 1: Mathematical Functions | ||
| Section 6 | Functions 2: Invertible Functions | ||
| Section 7 | Functions 3: O-Notation | ||
| Section 8 | Graph Theory | ||
| Section 9 | Trees, Binary Search Trees | ||
| Section 10 | Digraphs and Relation | ||
| Section 11 | Sequences and Series, with Telescoping Series | ||
| Section 12 | Proof by Induction, Convergence and Ratio Tests | ||
| Section 13 | Probability and Counting Problems | ||
| Section 14 | Matrices and Systems of Linear Equations |
[Domain] [Co-domain] [Image] [Ancestor] [Range]
[Constants] \((P_0)\) [Linear Functions] \((P_1)\) [Quadratic Functions] \((P_2)\) [Cubic Functions] \((P_3)\)
| ### Ordered Pairs |
| https://math24.net/cartesian-product-sets.html |
| ### Ordered Pairs In sets, the order of elements is not important. For example, the sets {2, 3} and {3, 2} are equal to each other. However, there are many instances in mathematics where the order of elements is essential. So, for example, the pairs of numbers with coordinates (2, 3) and (3, 2) represent different points on the plane. This leads to the concept of ordered pairs. |
| An ordered pair is defined as a set of two objects together with an order associated with them. Ordered pairs are usually written in parentheses (as opposed to curly braces, which are used for writing sets). |
| In the ordered pair the element is called the first entry or first component, and is called the second entry or second component of the pair. |
| Two ordered pairs and are equal if and only if and In general, |
| The equality is possible only if |
| ### Tuples The concept of ordered pair can be extended to more than two elements. An ordered tuple is a set of objects together with an order associated with them. Tuples are usually denoted by The element is called the entry or component, and is called the length of the tuple. |
| Similarly to ordered pairs, the order in which elements appear in a tuple is important. Two tuples of the same length and are said to be equal if and only if for all So the following tuples are not equal to each other: |
| Unlike sets, tuples may contain a certain element more than once: |
| Ordered pairs are sometimes referred as tuples. |
| ### Cartesian Product of Two Sets Suppose that and are non-empty sets. The Cartesian product of two sets and denoted is the set of all possible ordered pairs where and |
| The Cartesian product is also known as the cross product. |
| The figure below shows the Cartesian product of the sets and |
| Cartesian product of two sets A={1,2,3} and B={x,y}. Figure 1. It consists of ordered pairs: |
| Similarly, we can find the Cartesian product |
| As you can see from this example, the Cartesian products and do not contain exactly the same ordered pairs. So, in general, |
| If then is called the Cartesian square of the set and is denoted by |
| #### Cartesian Product of Several Sets Cartesian products may also be defined on more than two sets. |
| Let be non-empty sets. The Cartesian product is defined as the set of all possible ordered tuples where and |
| If then is called the Cartesian power of the set and is denoted by |
| #### Properties |
| Some Properties of Cartesian Product The Cartesian product is non-commutative: if only if either or The Cartesian product is non-associative: Distributive property over set intersection: Distributive property over set union: Distributive property over set difference: If then for any set Cardinality of Cartesian Product The сardinality of a Cartesian product of two sets is equal to the product of the cardinalities of the sets: |
| Similarly, |
| See solved problems on Page 2. |
| ### Videos * Functions (2005 Q4 Part A) * Functions (2003 Q4 Part B) * Floor Functions Exercise HD |
| ### Cartesian Product |
| \(X \times Y\) |
| \(X = \{1,2,3\}\) |
| \(Y = \{a,b,c\}\) |
| \(X \times Y = \{ (1,a) , (1,b) , (1,c) , (2,a), (2,b) ,(2,c), (3,a),(3,b),(3,c) \}\) \(Y \times X = \{ (a,1) , (a,2) , (a,3) , (b,1), (b,2) ,(b,3), (c,1),(c,2),(c,3) \}\) |
| Invertible Functions {data-navmenu=“Invertible Functions”} ===================================== |
| Column {.tabset} |
Necessary Conditions for Invertibility of a Function
The function must be one-to-one
The fucntion must be onto.
A function is onto if is range is equal to its codomain.
A function is one-to-one if no two distinct elements of the domain have the same image.