Finite and infinite sets
A finite set is one with a finite number of elements. For example, {1, 2, 7, 10} is a finite set as it has four elements. An infinite set is one that has an infinite number of elements such as ℝ, the set of real numbers. It is impossible to count how many real numbers there are. A countably infinite set is one that can be counted off by the natural numbers. Rational numbers are countably infinite in this way. However, German mathematician Georg Cantor proved that real numbers are not countable.
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Worked examples a For each of the following, explain which set(s) of numbers they belong to (either natural, integer, rational, irrational or real). A number may be a member of more than one set. i. 35 ■ Because this is a whole number (with no decimal places), this is a natural number and also an integer. ■ Because it can be written in the form , it is also a rational number. ■ Finally, it is also a real number. ii. 12.5 ■ The decimal point means that this is definitely not a natural number or an integer. , so is therefore a rational number. ■ It can be written in the form 25/2, therefore a rational number ■ In addition, it is also a real number. iii. √3 ■ In decimal form, this is 1.4142135… It does not terminate or recur, therefore it is an irrational number. ■ Just like every other number, it is also a real number. b Decide whether each of the following statements is true or false. i All natural numbers are also integers. This is TRUE. The set of integers includes all natural numbers AND also includes negative numbers. ii A square root is always irrational. This is FALSE. Most roots (such as 2 or 3 ) are irrational. However, 4 = 2 and 9 = 3. These are certainly not irrational! iii A number is either rational or irrational, but never both. This is TRUE. The definitions of these number sets are opposites: if a number is rational then it cannot be irrational and vice versa. Real numbers
Real numbers are the set of all possible real world quantities. Another way of thinking about real numbers is that they are the points on an infinitely long line. Every possible value, both positive and negative, appears somewhere on the line. The symbol ℝ is used for this set. ℝ contains all natural numbers (ℕ), integers (ℤ), rational numbers (ℚ) and irrational numbers. Real numbers are very useful for measuring. For example, if three people ran a race, their times (in seconds) may be 15, 17.3 and 18.3454. These are all real numbers. Irrational numbers
Irrational numbers are those numbers which are not rational: that is, they cannot be written as a fraction. Famous examples of irrational numbers include π, e and √2. The decimal version of any irrational number neither terminates nor recurs. For example, π can be written as 3.141592… but this series of digits will continue infinitely. The world record for the number of digits calculated for π currently stands (as of 2015) at 2.7 trillion digits. For this reason, any computer representation of an irrational number will actually be a very close approximation. Rational Numbers
Rational numbers are those numbers that can be written as a fraction, such as s 7/5 or 3/2. The symbol ℚ is used for the set of rational numbers and can be thought of as standing for ‘quotient’ to aid remembering. A rational number can be defined as a number in the form of x/y where x and y are both members of ℤ and y is not zero. Since a number such as 9 can be written as the fraction 9/1 ℚ includes all members of ℤ. When any rational number is written as a decimal, it either terminates after a finite number of digits (such as 1/8= 0.125) or or recurs (such as 1/3 = 0.333333.....) Integers
Integers are negative and positive whole numbers, again including 0. The symbol ℤ is used for the set of integers, so that ℤ = {…, –3, –2, –1, 0, 1, 2, 3, …}. ℤ is used for integers because ‘Zahlen’ is German for ‘numbers’. Just as with natural numbers, if a member of ℤ is added to or multiplied by another member of ℤ, the result is also a member of ℤ. This is also true for subtraction as ℤ includes the negative whole numbers. Integers are therefore very useful as counters in computer programs for whole quantities that may be negative. Natural numbers
Natural numbers are positive whole numbers, including 0. The symbol ℕ is used for the set of natural numbers, so that ℕ = {0, 1, 2, 3, …} The natural numbers are used for counting (such as ‘there are 17 students in this class’) and ordering (such as ‘the team finished in 3rd place’). In computer science, it is important to remember that, if a member of ℕ is added to or multiplied by another member of ℕ, the result is also a member of ℕ. This allows natural numbers to be used as counters in computer programs. Abiblo Computer Science - Acceleration Time
Acceleration time also known as start time or the time taken for device to reach its speed of operation from the quiescent state. Abiblo Computer Science - Abstract Computability Theory
It is a theory of function which could be computed on any algebra by algorithms. The aim of abstract computability theory is to explore the scope and limitation of computation of any data. It is a generalisation to arbitrary many sorted algebras of the theory of the calculable or recursive function on the natural numbers. Abstract computability theory begins with classification and analysis of many models of computation and specification which apply to algebras. This lead to Church - Turing thesis which establishes , function on an abstract data type which are programmable by a deterministic programming language. Comparison is possible between computations on modelling data types, algebras and implementations. There will be a new theories of computation for special data types arrived from the theory such as algebras of real numbers. The programming language is a form of simple example of a method for computing functions on many sorted algebra A. It can compute all partial recursive function on natural numbers. Computation focuses on the operation of algebras ( sequencing, iteration and branching which available as limited means of searching A ). However, a vital missing component is the capacity to compute with finite sequence of data from A, Pairing functions can stimulates the natural number of finite sequences but not possible to stimulate finite sequence on algebra A. A new algebra A * is created by addition of finite sequence and operations for every data set in A. Most of the results derived from the theory of computability on the natural number can also be proven for abstract computability theory on finite generated minimal algebra. |
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