Language-of-Sets/ Mathematics in the Modern World
- 2. SET • Sets are represented as a collection of well-defined objects or elements and it does not change from person to person. • A set is represented by a capital letter and its elements are denoted by small letters. • The number of elements in the finite set is known as the cardinal number of a set.
- 3. What are the Elements of a Set? • Let us take an example: A = {1, 2, 3, 4, 5 } • Since a set is usually represented by the capital letter. • Thus, A is the set and 1, 2, 3, 4, 5 are the elements of the set or members of the set. • The elements that are written in the set can be in any order but cannot be repeated. • All the set elements are represented in small letter in case of alphabets. • Also, we can write it as 1 A, 2 A etc. ∈ ∈ • Some commonly used sets are as follows: • N: Set of all natural numbers • Z: Set of all integers • Q: Set of all rational numbers • R: Set of all real numbers • Z+ : Set of all positive integers
- 4. Natural Numbers (N) • Are the numbers that start from 1 and end at infinity. • In other words, natural numbers are counting numbers and they do not include 0 or any negative or fractional numbers. For example: 1, 6, 89, 345, and so on INTEGERS (Z) - An integer is a number with no decimal or fractional part and it includes negative and positive numbers, including zero.
- 5. RATIONAL NUMBERS (Q) • In addition to all the fractions, the set of rational numbers includes all the integers. • Can be written as a quotient with the integer as the numerator and 1 as the denominator. • is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero. • In decimal form, rational numbers are either terminating or repeating decimals.
- 6. REAL NUMBERS (R) • It includes rational numbers like positive and negative integers, fractions, and irrational numbers. • In other words, any number that we can think of, except complex numbers, is a real number. • For example: 3, 0, 1.5, 3/2, √5, and so on are real numbers. COMPLEX NUMBERS • The numbers that are expressed in the form of a+ib where, a,b are real numbers and 'i' is an imaginary number called “iota”. • The value of i = (). • For example: 2+3i is a complex number, where 2 is a real number (Re) and 3i is an imaginary number (Im).
- 7. A. Representation of a Set 1. Using the Set-Roster Notation - If S is a set, the notation x S means that x is an element of S. ϵ - The notation x S ∉ means that x is not an element of S. a set may be specified using the set roster notation by writing all of its elements between curly braces. • For example, {1,2,3} denotes the set whose elements are 1, 2, and 3. - A variation of the notation is sometimes used to describe a very large set, as when we write {1,2,3,…, 100} to refer to the set of all integers from 1 to 100. - A similar notation can also describe an infinite set, as when we write {1,2,3,…} to refer to the set of all positive integers. (The symbol … is called an ellipsis and is read “and so forth.”)
- 9. Examples: Note: To write a set in Roster form elements are not to be repeated. RULE METHOD: 1. V= {x:x set of vowels in the word PROPOSITION}. 2. If A be the set of natural numbers less than 7. 3. If A be the set of letters used in the word mathematics. 4. C={x|x N and 50≤x<60} 5. D = {x|x R and x2 – 5x + 6 =0} SET ROSTER NOTATION: 1. V = { i, o,} 2. A={1, 2, 3, 4, 5, 6}. 3. A = {m, a, t, h, e, i, c, s} 4. C={50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60} 5. x2 – 5x + 6 =0 (x-3)(x-2)=0 x= 3, 2 D={2, 3} QUESTION ANSWERS
- 10. TYPES OF SET Finite Set • A finite set has a finite or countable number of elements. • It has limited set. • It is a set of natural numbers i.e., positive integers and can be easily counted. • It is expressed as P = {1, 2, 3, 4, . . ., n} for natural number n. For example: 1. A= {5, 6, 7, 8} 2. P= {x:x set of vowels in the English alphabet} Note: An empty set { } is also considered a finite set as it has zero elements Example: P={ } or n(A) = 0.
- 11. Infinite Set • It has an infinite or uncountable number of elements. • It has unlimited set. • It is a set of all whole numbers including non-negative and negative integers. Infinite sets are also known as uncountable sets. Example: 1. P = {0, 1, 2, 3, . . . }. 2. P= {x:x set of whole numbers} 3. P= {x:x set of white stones in the river}
- 13. UNIT SET • A set that has only one element is called a singleton set. • It is also known as a unit set because it has only one element. Example: Set A = { k|k is an integer between 5 and 7} = {6}. The answer is only one. No other answer except 6. Empty or Null Sets • A set that does not contain any element is called an empty set or a null set. • An empty set is denoted using the symbol ' '. ∅ It is read as 'phi'. Example: 1. Set X = {}. 2. A= {set of Dinasaurs in Bontoc}
- 14. LET’S TRY: A. write the following in Set Roster Notation. 1. The set of consonants in the word ‘possession’. 2. The set of first three letters in the word ‘booklet’. 3. {x : x is a letter in the word ‘SCHOOL’}. 4. {x : x is an odd natural number between 10 and 20}. 5. {Vowels used in the word ‘AMERICA’} B. Write the following set in the set builder form: 1. A = { 2, 4, 6, 8, 10, 12, 14}. 2. 0= {11, 13, 15, 17, 19} 3. P= {2, 3, 5, 7, 11, 13} 4. S= {Math, Science, History, Filipino, English} 5. B= {Omfeg, Loc-og, Samoki, Eyeb, Chakchakan, Tocucan, Lanao}
- 15. Identify the types of sets. 1. T= {x:x set of Math subjects offered in MPSU-Tadian} and B = {x:x set of Math subjects offered in MPSU-Bontoc} 2. {1,3,5, . . . } 3. {2, 3, 4, 5} 4. {x:x set of integers} 5. M={} 6. {x:x set of apples in the basket of grapes} 7. {x:x the president of MPSU} 8. If A = {1,2,3,4} and B = {Red, Blue, Green, Black} 9. {x:x x 10} 10. {x: x 97 < x > 99}