Relations and Functions. In mathematics, relations and functions describe how one set of numbers connects to another. They form the foundation of algebra and are essential for understanding graphs, equations, and real-life mathematical models. What Is a Relation? A relation is a rule or connection that pairs elements from one set (called the domain ) with elements from another set (called the range ). If we have two sets: Set A = {1, 2, 3} Set B = {4, 5, 6} A relation from A to B can be written as a set of ordered pairs such as: R = { ( 1 , 4 ) , ( 2 , 5 ) , ( 3 , 6 ) } R = \{(1, 4), (2, 5), (3, 6)\} This means: 1 is related to 4 2 is related to 5 3 is related to 6 Each pair shows how an element from the first set connects to one from the second. Ways of Representing a Relation Relations can be represented in several forms: Set of Ordered Pairs: e.g., R = { ( 2 , 3 ) , ( 3 , 4 ) , ( 4 , 5 ) } Mapping Diagram: uses arrows to show how each el...
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Number Bases. Numbers are the foundation of all mathematics, but did you know that the way we represent them depends on the base we use? The base of a number system tells us how many digits are used before the digits start repeating in patterns. The most common base we use in everyday life is Base 10 , also called the Decimal System . However, there are many other bases such as Binary (Base 2) , Octal (Base 8) , and Hexadecimal (Base 16) — all widely used in computing and digital systems. What Is a Number Base? A number base (or radix ) is the number of unique digits used to represent numbers in a positional number system. For example: In Base 10 , we use digits 0 to 9. In Base 2 , we use digits 0 and 1 only. In Base 8 , we use digits 0 to 7. In Base 16 , we use digits 0 to 9 and letters A to F (representing values 10 to 15). So, the value of ea...
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Variation. In mathematics, variation explains how one quantity changes in relation to another. It helps us understand relationships like how time affects distance, or how speed affects fuel usage. There are four main types of variation: direct , inverse , joint , and partial (combined) variation. Let’s go through each type with detailed examples. 1. Direct Variation In direct variation , one quantity increases (or decreases) as the other increases (or decreases). If y varies directly as x , the relationship is: y = k x where k k is the constant of variation . Example 1 If y y varies directly as x x , and y = 10 y = 10 when x = 5 x = 5 , find y y when x = 8 x = 8 . Solution: y = k x y = kx Substitute y = 10 y = 10 , x = 5 x = 5 : 10 = 5 k ⇒ k = 2 10 = 5k \Rightarrow k = 2 When x = 8: y = 2 ( 8 ) =...