Chapter 2: Mathematical Language and Symbols PDF

GE 3/GE 4: Mathematics in the Modern World 7 Chapter Mathematical Language and Symbols 2 LESSON 3: SETS Learning Objectives At the end of this module, you are expecte...

GE 3/GE 4: Mathematics in the Modern World 7 Chapter Mathematical Language and Symbols 2 LESSON 3: SETS Learning Objectives At the end of this module, you are expected to: a. define set; and b. familiarize symbols about set. Take these challenge. Think of a word that relates to the word “set” and write it to each of the quadrilaterals. Set 1. Based on the activity, what is a set? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 2. In your opinion, how can the word 'set' be used in the field of mathematics? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ CONCEPT NOTES The word ‘set’ is one of the words with the most meaning in the English Dictionary. According to Oxford English Dictionary, the word ‘set’ holds 430 definitions. The use of the word set as formal mathematical term was introduced in 1879 by Georg Cantor (1845-1918). For most mathematical purposes we can think of set intuitively, as Cantor did, simply as collections of elements. Georg Cantor is known as the Father of Set Theory.  GE 3/GE 4: Mathematics in the Modern World 8 Georg Ferdinand Cantor, although he was born in St. Petersburg and lived there until 1856, should properly be ranked among the German mathematicians, because he was educated and employed in German universities. His stockbroker father had urged him to study engineering, a more profitable pursuit in mathematics, and with this intention Cantor began his university studies at Zurich in 1862. The elder Cantor finally agreed to allow his son to follow career in mathematics, so that after a semester at Zurich he moved to the University of Berlin. There he attended the lectures of the great triumvirate, Weierstrass, Kummer, and Kronecker. In 1867, he received his Ph.D. from Berlin, having submitted a thesis on problems in number theory, a thesis that in no way foreshadowed his future work. Two years later, Cantor accepted an appointment as privatdozent at Hale University, where he remained until his retirement in 1913. Sets is a well-defined collection of distinct objects, things, persons, places and etc. Sets are usually denoted by capital letters such as A, B, C and so on and enclosed with curly brackets or braces { }. The symbols "∈" and "∉" are used to indicate that an object is or is not an element of a set respectively. For example, if S represents the set of all flowers, then rose ∈ S and mango ∉ S. LESSON 4: THE REAL NUMBER SYSTEM Learning Objectives At the end of this module, you are expected to: a. recall the Real Number System; and b. familiarize symbols about used in Real Number System. Recall the Real Number System. Illustrate it using a Venn Diagram. Questions: 1. Which set of number does 0 (zero) belongs? ____________________ 2. Based on the Venn diagram, what are the elements of a set of integer? set of whole number? __________________________________________________________________________________ _________________________________________________________________________________ CONCEPT NOTES The Real Number System 1. The sets of Natural or Counting numbers (N) contains all positive integers or the elements { 1, 2, 3, …. +ꝏ}. 2. The sets of Whole Number (W) includes the elements of the natural number plus zero (0). 3. The sets of Integers (Z) includes all the elements of the set of whole numbers and the negative integers.  GE 3/GE 4: Mathematics in the Modern World 9 4. The sets of Rational Number (Q) includes all numbers that can be written as a fraction or as a ratio of integers. However, the denominator cannot be equal to zero. It may appear in the form of decimal. If a decimal number is repeating or terminating, it can be written as a fraction, therefore, it must be rational number. 5. The sets of Irrational Number (Q-) are numbers that cannot be written as a ratio of two integers. These are the leftover numbers after all rational numbers are removed from the set of the real numbers. If written in decimal form, it don’t terminate and don’t repeat. THE REAL NUMBER SYSTEM Real Number (R) Rational Number (Q) Irrational Number (Q-) Fractions Integers (Z) Negative Integer (Z-) Whole Number (W) Zero NaturaL Number (N) Positive Number (Z+) LESSON 5: THREE METHODS OF WRITING A SET Learning Objectives At the end of this module, you are expected to: a. memorize the methods of writing a set; and b. write sets in different methods namely: description method, set roster notation, and set-builder notation. CONCEPT NOTES 1. Description Method It is in a form of words describing what is included in a set. Example: Set A is the set of primary colors. 2. Set Roster Notation or listing of elements If S is a set, the notation x Є S means that x is an element of S. The notation x Є 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 element between 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 we write {1, 2, 3, … , 100} to refer to the  GE 3/GE 4: Mathematics in the Modern World 10 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”.) Example: A = { red, yellow, blue} 3. Set-builder Notation Let S denote a set and let P(x) be a property that elements of S may or may not satisfy. We may define a new set to be the set of all elements x in S such that P(x) is true. We denote this set as { x Є S І P(x)}. Example: A = { x І x Є of primary color}* *This reads, Set A is the set of all elements x such that x is a primary color. LESSON 6: TYPES OF SET Learning Objectives At the end of this module, you are expected to: a. identify the different types of set; and b. solve problems involving the types of set. CONCEPT NOTES 1. Single/singleton set Definition: If a set contains only one element it is called single/singleton set. Hence, the set given by {1}, {0}, {a} are all consisting of only one element and therefore are single/singleton sets. Examples: a. Set A is a set of even prime number. b. B = {xІx Є N, 2𝑥𝑥 2 − 6 = 𝑥𝑥} B = {2} 2. Finite set Definition: A set consisting of a natural number of objects, i.e. in which number of element is finite is said to be a finite set. It is a set which consists of a definite number of elements or the elements can be counted. Examples: a. Set A is a set of days in a week. b. C = {xІx Є N, 2 < x < 8} C = {3, 4, 5, 6, 7} c. D = {xІx Є Z, 𝑥𝑥 2 − 26 = −10 } D = {-4, 4} 3. Infinite set Definition: If the number of elements in a set is cannot be counted, the set is said to be infinite set. Ellipses {…} will be used to denote infinity. Examples: a. Set A is the set of all integers more than -3. b. Set B is the set counting numbers. c. C = {xІx Є Z, x ≤ 3} C = {- ꝏ…, 0, 1, 2, 3}  GE 3/GE 4: Mathematics in the Modern World 11 d. D = {xІx Є N and x is odd} D = {1, 3, 5, 7, … , +ꝏ} 4. Null set/Empty Set Definition: A set which does not contain any element is called null/empty set. It is denoted by ø or { }. Examples: a. A = {xІ1 < x < 2 and x is a natural number} b. B = {xІx2 – 2 = 0 and x is a rational number} c. C = {xІx is an even prime number greater than 2} d. D = {xІx2 = 4, x is odd} Note: An empty set is also a finite set. 5. Subset Definition: If two sets A and B are such that every element of A is also an element of B then we say that A is a subset of B. It is denoted by A ⊆ B. A subset A is said to be subset of B if every elements which belongs to A also belongs to B. Example: A = { 1, 2, 3} B = { 1, 2, 3, 4} A is a subset of B, denoted by A ⊆ B The number of subsets of a set is 2n where n is the number of elements in a given set. Thus, set A has 3 elements, 23 = 8 subsets and set B has 4 elements, 24 = 16 subsets. Note: An empty set is always a subset of any given set. 6. Power set Definition: Power set of a set is defined as a set of every possible subset. Example: If A = {2, 3, 4, 5}, then P(S) = { ø, {2}, {3}, {4}, {5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}, {2, 3, 4, 5} }. 7. Proper Subset Definition: Let A and B be sets. A is a proper subset of B if, and only if, every element of B is in A but there is at least one element of B that is not in A. Example: If A = {1, 3} and B = {1, 2, 3}, then, A is a proper subset of B, denoted by A ⊂ B. 8. Superset Definition: Set A is a superset of set B if all elements of set B are elements of the set A. Example: If A = {1, 2, 3, 4} and B = { 1, 2, 4}, then, A is a superset of B, denoted by A⊃B. 9. Universal Set Definition: Any set which is a superset of all the sets under consideration is said to be universal set and is either denoted by omega (S) or U. 10. Joint sets Definition: Let A and B be sets. If sets A and B has shared common element/s, then these sets are joint sets. Example: A = {1, 2, 3} and B = {1, 2, 3, 4, 5}, then sets A and B are joint sets.  GE 3/GE 4: Mathematics in the Modern World 12 11. Disjoint sets Definition: If two sets A and B should have no common elements or we can say that the intersection of any two sets A and B is the empty set, then these sets are known as disjoint sets. Example: A = {1, 2, 3} and B = {4, 5}, then sets A and B are disjoint sets. 12. Equivalent sets – two or more sets having the same number of elements Definition: Two finite sets with an equal number of elements are called equivalent sets. If sets A and B are equivalents, we write A⟷B. Example: A = {1, 2, 3} and B = {a, b, c}, then sets A and B are equivalent sets. 13. Equal sets – if both sets have the same element contain Definition: Two sets are said to be equal or identical to each other, if they contain the same element. If an element of set A is the same with set B, then A = B. Example: A = {1, 2, 3} and B = {1, 2, 3}, then sets A and B are equal sets. LESSON 7: SET OPERATIONS Learning Objectives At the end of this module, you are expected to: a. recognize the set operations, and b. solve problems involving set operations. CONCEPT NOTES 1. The union of two sets A and B is a set containing all elements that are in set A or in set B. It is denoted by 𝐴𝐴 ∪ 𝐵𝐵. 2. The intersection of two sets A and B, denoted by 𝐴𝐴 ∩ 𝐵𝐵, consist of all elements that are both in set A and B. 3. The complement of set A, denoted by A’, is the set of all elements that are in universal set but are not in set A. 4. The difference (subtraction) of set A and B, denoted by A – B, consist of elements that are in set A but not in set B. Note: A – B is not the same as B – A. 5. A Cartesian Product of two sets A and B, written as A X B, is the set containing ordered pairs from set A and B. That is, if C = A X B, then each element of C is of the form (x, y), where x in set A and y in set B. Note: A X B is not the same as B X A. EXAMPLE: Perform the indicated operations. Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} A= {1, 2, 4, 6} B= {3, 4, 7} C= {1, 3, 5, 7} D= {2, 4, 6, 8} {(3,1),(3,3),(3,5),(3,7),(4,1),(4,3),(4 1. A’ = {3, 5, 7, 8, 9, 10, 11, 12} 11. BxC = ,5),(4,7), (7,1),(7,3),(7,5),(7,7)} 2. C’ = {2, 4, 6, 8, 9, 10, 11, 12} 12. (B ∩ C) - (A - C) = {3, 7}  GE 3/GE 4: Mathematics in the Modern World 13 3. A∩C = {1} 13. (A U B) ∩ B’ = {1, 2, 6} 4. A∩B = {4} 14. C’ – A’ = {2, 4, 6} 5. B–D = {3, 7} 15. (B – A)’ U (C – A) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} 6. A–B = {1, 2, 6} 16. D’– (A U D) = {3, 5, 7, 9, 10, 11, 12} 7. BUC = {1, 3, 4, 5, 7} 17. (A’- C’) ∩ (B’ ∩ D’) = {5} {(1,1),(1,5),(3,1),(3,5),(5,1),(5,5),(7 8. A’∩ B’ = {5, 8, 9, 10, 11, 12} 18. (C – D) x (C – B) = ,1),(7,5)} A’ U 9. = {8, 9, 10, 11, 12} 19. A – B’ = {4} C’ {(1,3),(1,4),(1,7),(2,3),(2,4), 10. A x B = (2,7),(4,3),(4,4),(4,7),(6,3),(6,4) 20. (C’ – B) – (A ∩ D) = {8, 9, 10, 11, 12} ,(6,7)}