Chapter 3 - The Language of Sets PDF

course outline (GE Math 1) Chapter 1 – The Nature of Mathematics Chapter 2 – Mathematical Language and Symbols Chapter 3 – The Language of Sets Chapter 4 – Functions and Relations Chapter 5 – Logic and Conditional Statements Chapter 6 – Patterns and Problem Solving Chapter 7 – Statisti...

course outline (GE Math 1) Chapter 1 – The Nature of Mathematics Chapter 2 – Mathematical Language and Symbols Chapter 3 – The Language of Sets Chapter 4 – Functions and Relations Chapter 5 – Logic and Conditional Statements Chapter 6 – Patterns and Problem Solving Chapter 7 – Statistics Chapter 8 – Graph Theory Chapter 9 – Modular Arithmetic class rules ❑ RULE #1: ❑ RULE #3: ❑ RULE #2: ❑ RULE #4: CHAPTER 3. MATHEMATICAL LANGUAGE AND SYMBOLS – Language of Sets Core Idea “Mathematics has its own symbols, syntax, and rules.” learning objectives 1. discuss the definitions of a set, subset, and proper subset; 2. determine whether a set is in roster or set- builder notation; 3. find all possible subsets of a given set; 4. perform set operations; and 5. illustrate subset and universal set using Venn Diagram. SET It is a collection of related and well-defined objects called elements (denoted by ∈). 𝐶 = {𝑥|0 < 𝑥 < 5} 𝐴 = {1, 2, 3, 4, 5, 6, 7, 8, 9} 𝐶 = {1, 2, 3, 4} 𝐵 = {𝑟𝑒𝑑, 𝑦𝑒𝑙𝑙𝑜𝑤, 𝑏𝑙𝑢𝑒} Historical background Georg Cantor (1845-1918) introduced the word set in 1879. UNIVERSAL SET It is collection of all elements of all the related sets, known as its subsets. The universal set is usually denoted by 𝑼. Symbol Set ℝ real numbers ℝ+ positive real number ℤ all integers ℤ+ positive integers ℚ rational numbers ℕ natural numbers WRITING A SET ❑ Roster Notation is a way of listing the elements separated by a comma. Example: Let 𝐴 and 𝐵 be sets. If 𝐴 is the set of all even whole numbers between 1 and 10, and 𝐵 is the set of all odd whole numbers between 1 and 10 write set 𝐴 and 𝐵 in roster notation. Solution: 𝐴 = {2, 4, 6, 8} ; 𝐵 = {3, 5, 7, 9} WRITING A SET ❑ Set-Builder Notation it is a way of representing or explaining the properties that must satisfy by the elements of a set. Example: Consider 𝑋 and 𝑌 as sets of natural numbers. Let 𝑋 = {1, 2, 4, 3,5, 6, 7} and 𝑌 = 11,12,13,14. Write the sets in set-builder notation. Solution: 𝑋 = {𝑥 ∈ ℕ|0 < 𝑥 < 8} 𝑌 = {𝑦 ∈ ℕ|10 < 𝑦 < 15}  It is a set which contains no elements. It is EMPTY SET usually denoted as { } or ∅. The empty set is always considered a subset of any set. Is {0} a null set? No. CARDINALITY OF A SET Let 𝐴 be a set. The cardinality of a set, denoted by 𝑛(𝐴) is the number of elements in 𝐴. Let 𝐴 = {2,4,6,8,10}, then 𝑛(𝐴) = 5.  Equal sets are sets whose elements are exactly EQUAL SETS the same, otherwise, the sets are unequal. EQUIVALENT SETS Equivalent sets are sets having the same cardinal number, otherwise they are non-equivalent sets. Examples: Given the following sets: P = {x/x is a letter from the word “taste”}. R = {x/x is a letter from the word “eats”}. S = {x/x is a letter from the word “test”}. Which two sets are equal? Which sets are equivalent? SUBSET These are sets contained in a universal set or another set. Definition. If 𝐴 and 𝐵 are sets, then 𝐴 is called a subset of 𝐵, denoted by 𝐴 ⊆ 𝐵, if and only if, every element of 𝐴 is also an element of 𝐵. Symbolically, 𝐴 ⊆ 𝐵 means that for all elements 𝑥 ∈ 𝐴, then x ∈ 𝐵. 𝐴 ⊆ 𝐵 is read as “𝐴” is a subset of “𝐵”. 𝐴 ⊈ 𝐵 is read as “𝐴” is not a subset of “𝐵”. SUBSET These are sets contained in a universal set or another set. Example: Find the subsets of the following sets. a. P = 𝑠, 𝑢, 𝑛 b. Q = 𝑠, 𝑒, 𝑎, 𝑙 c. S = 𝑔, 𝑟, 𝑒, 𝑎, 𝑡 SUBSET These are sets contained in a universal set or another set. Solution: Find the subsets of the following sets. a. P = 𝑠, 𝑢, 𝑛 , 𝑠 , 𝑢 , 𝑛 , 𝑠, 𝑢 , 𝑠, 𝑛 , 𝑢, 𝑛 , {𝑠, 𝑢, 𝑛} , 𝑠 , 𝑒 , 𝑎 , 𝑙 , 𝑠, 𝑒 , 𝑠, 𝑎 , 𝑠, 𝑙 , 𝑒, 𝑎 , b. Q = 𝑠, 𝑒, 𝑎, 𝑙 𝑒, 𝑙 , 𝑎, 𝑙 , 𝑠, 𝑒, 𝑎 , 𝑠, 𝑒, 𝑙 , 𝑠, 𝑎, 𝑙 , 𝑒, 𝑎, 𝑙 , {𝑠, 𝑒, 𝑎, 𝑙} c. S = 𝑔, 𝑟, 𝑒, 𝑎, 𝑡 How many subsets can be obtained here without enumeration? NUMBER OF SUBSETS How many subsets are there if the sets has: 1 element = 2 subsets Thus, to get the number 2 elements = 4 subsets of subsets of a given set, use the formula 𝟐𝒏. 3 elements = 8 subsets For example, if 𝒏 𝑨 = 𝟔 4 elements = 16 subsets elements, then 𝟐𝟔 = 𝟔𝟒 subsets.  Definition. PROPER SUBSET Let 𝐴 and 𝐵 be sets. 𝐴 is a proper is any subset of a set subset of 𝐵, if and only if, every except itself. element of 𝐴 is in 𝐵 but there is at least one element of 𝐵 that is not in 𝐴. 𝐴 ⊂ 𝐵 is read as “𝐴” is a proper subset of “𝐵”. Example: Let 𝐴 = 1,4,3 ; 𝐵 = 1,2,3,4,5 ; 𝐶 = {2,3,1,4,5}. Is 𝐴 ⊂ 𝐵? YES. Is 𝐶 ⊂ 𝐵? No. Is C ⊆ 𝐵? YES. POWER OF A SET It is the set of all subsets for any given set which includes the empty set. Example: Consider 𝑆 = {𝑎, 𝑖, 𝑟. Let 𝑃(𝑆) denotes the power of a set 𝑆. So, P( S ) =   , a, r,  i , a, i, a, r, i, r, i, r , a  Based on 𝑆 and 𝑃(𝑆) tell whether each of the following is TRUE or FALSE. a  P(S ) a P(S ) a  P(S ) a, r  P( S ) EXERCISES Determine whether the statement is true or false. 1. 𝑥 ∈ 𝑥, 𝑦, 𝑧 2. 𝑦 ⊆ 𝑥, 𝑦, 𝑧 3. 𝑥 ∈ 𝑥, 𝑦, 𝑧 4. {𝑦} ⊆ 𝑥, 𝑦, 𝑧 5. {𝑥} ⊆ 𝑥𝑦 6. {𝑎, 𝑏, 𝑐, 𝑑} has 12 subsets OPERATIONS OF SETS ❑ Union ❑ Intersection ❑ Difference ❑ Complement ❑ Cartesian Product  The union of two or more sets contains UNION OF SETS ALL the elements in all the sets under consideration. Suppose 𝐴 and 𝐵 are sets. The union of sets 𝐴 and 𝐵 is denoted by 𝑨 ∪ 𝑩. Example: Consider 𝐴 = 1,2,3,4,5,6 and 𝐵 = 2,4,6,8,10,12. Find 𝐴 ∪ 𝐵. INTERSECTION OF SETS The intersection of two or more sets contains the common elements in all the sets under consideration. Suppose 𝐴 and 𝐵 are sets. The intersection of sets 𝐴 and 𝐵 is denoted by 𝑨 ∩ 𝑩. Example: Consider 𝐴 = 1,2,3,4,5,6 and 𝐵 = 2,4,6,8,10,12. Find 𝐴 ∩ 𝐵. COMPLEMENT OF A SET The complement of a set is all of the elements in the universal set 𝑈 not found in the considered set. Suppose 𝐴 is a set. The complement of set 𝐴 is denoted by 𝑨′. Example 𝑈 = {1,2,3,4,5,6,7,8,9,10} Consider U = 𝑥 ∈ ℕ|1 ≤ 𝑥 ≤ 10 and A = {2,4,6,8}. Find 𝐴′. DIFFERENCE OF SETS Suppose 𝐴 and 𝐵 are sets. The difference of 𝐴 by 𝐵, denoted by 𝐴 − 𝐵 is the set which contains the elements of 𝐴 excluding the elements found in both 𝐴 and 𝐵 and vice versa for the difference of 𝐵 by 𝐴 denoted by 𝐵 − 𝐴. Example: Consider 𝐴 = 1,2,3,4,5,6 and 𝐵 = 2,4,6,8,10,12. Find 1 𝐴 − 𝐵; 2 𝐵 − 𝐴. Answers: (1) 𝐴 − 𝐵 = {1,3,5} (2) 𝐵 − 𝐴 = {8,10,12} CARTESIAN PRODUCTS Given sets 𝐴 and 𝐵. The Cartesian product of 𝐴 and 𝐵, denoted 𝐴 × 𝐵 and read “A cross B” is the set of all ordered pairs (𝑎, 𝑏) where 𝑎 ∈ 𝐵 and 𝑏 ∈ 𝐵. Symbolically, 𝐴 × 𝐵 = { 𝑎, 𝑏 𝑤ℎ𝑒𝑟𝑒 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵}. Example 10. Consider 𝐴 = 0,1,2 and 𝐵 = 1,4,5. Find 1 𝐴 × 𝐵; 2 𝐵 × 𝐴. Answers: (1) 𝐴 × 𝐵 = { 0,1 , 0,4 , 0,5 , 1,1 , 1,4 , 1,5 , 2,1 , 2,4 , (2,5)} (2) 𝐵 × 𝐴 = { 1,0 , 1,1 , 1,2 , 4,0 , 4,1 , 4,2 , 5,0 , 5,1 , (5,2)} VENN DIAGRAM A Venn diagram is an illustration that uses circles to show the relationships among finite groups of objects or elements of different finites sets. VENN DIAGRAM VENN DIAGRAM VENN DIAGRAM End of Discussion…