Ion Goian ■ Raisa Grigor ■ Vasile Marin ■ Florentin Smarandache ALGEBRAIC PROBLEMS AND EXERCISES FOR HIGH SCHOOL The Educational Publisher Columbus, 2015 Algebraic problems and exercises for high school Ion Goian ■ Raisa Grigor ■ Vasile Marin ■ Florentin Smarandache ALGEBRAIC PROBLEMS AND EXERCISES FOR HIGH SCHOOL Sets, sets operations ■ Relations, functions ■ Aspects of combinatorics 1 Ion Goian ■ Raisa Grigor ■ Vasile Marin ■ Florentin Smarandache First imprint: Algebra în exerciții și probleme pentru liceu (in Romanian), Cartier Publishing House, Kishinev, Moldova, 2000 Translation to English by Ana Maria Buzoianu (AdSumus Cultural and Scientific Society) The Educational Publisher Zip Publishing 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Email: [email protected] AdSumus Cultural and Scientific Society Editura Ferestre 13 Cantemir str. 410473 Oradea, Romania Email: [email protected] ISBN 978-1-59973-342-5 2 Algebraic problems and exercises for high school Ion Goian ■ Raisa Grigor ■ Vasile Marin ■ Florentin Smarandache ALGEBRAIC PROBLEMS AND EXERCISES FOR HIGH SCHOOL ■ Sets, sets operations ■ Relations, functions ■ Aspects of combinatorics The Educational Publisher Columbus, 2015 3 Ion Goian ■ Raisa Grigor ■ Vasile Marin ■ Florentin Smarandache © Ion Goian, Raisa Grigor, Vasile Marin, Florentin Smarandache, and the Publisher, 2015. 4 Algebraic problems and exercises for high school Foreword In this book, you will find algebraic exercises and problems, grouped by chapters, intended for higher grades in high schools or middle schools of general education. Its purpose is to facilitate training in mathematics for students in all high school categories, but can be equally helpful in a standalone work. The book can also be used as an extracurricular source, as the reader shall find enclosed important theorems and formulas, standard definitions and notions that are not always included in school textbooks. 5 Ion Goian ■ Raisa Grigor ■ Vasile Marin ■ Florentin Smarandache Contents Foreword / 5 Notations / 7 1. Sets. Operations with sets / 9 1.1. Definitions and notations / 9 1.2. Solved exercises / 16 1.3. Suggested exercises / 32 2. Relations, functions / 43 2.1. Definitions and notations / 43 2.2. Solved exercises / 59 2.3. Suggested exercises / 86 3. Elements of combinatorics / 101 3.1. Permutations. Arrangements. Combinations. Newton’s binomial / 101 3.2. Solved problems / 105 3.3. Suggested exercises / 125 Answers / 137 References / 144 6 Algebraic problems and exercises for high school Notations = equality; ≠ inequality; ∈ belongs to; ∉ doesn’t belong to; ⊆ subset; ⊇ superset; ∪ union; ∩ intersection; ∅ empty set; ∨ (or) disjunction; ∧ (and) conjunction; 푑푒푓 푝 ⇔ 푞 p equivalent with 푞; ℕ = {0,1,2,3,4,…} natural numbers ℤ = {…,−2,−1,0,1,2,…} integer numbers set; 푚 ℚ = { |푚,푛∈ℤ,푛≠0} rational numbers set; 푛 ℝ real numbers set; ℂ = {푎 + 푏|푎, 푏 ∈ 푅, 2 = −1} complex numbers set; 퐴+ = {푥 ∈ 퐴|푥 > 0}, 퐴 ∈ {ℤ, ℚ, ℝ}; 퐴− = {푦 ∈ 퐴|푦 < 0}, 퐴 ∈ {ℤ, ℚ, ℝ}; 퐴∗ = {푧 ∈ 퐴|푧 ≠ 0} = 퐴 ∖ {0}, 퐴 ∈ {ℕ, ℤ, ℚ, ℝ, ℂ}; |푥| absolute value of 푥 ∈ ℝ; [푥] integer part of x ∈ R; {푥} fractional part of 푥 ∈ ℝ, 0 ≤ {푥} < 1; (푎, 푏) pair with the first element 푎 and the second element 푏 (also called "ordered pair"); 7 Ion Goian ■ Raisa Grigor ■ Vasile Marin ■ Florentin Smarandache (푎, 푏, 푐) triplet with corresponding elements 푎, 푏, 푐; 퐴 × 퐵 = {푎, 푏|푎 ∈ 퐴, 푏 ∈ 퐵} Cartesian product of set 퐴 and set 퐵; 퐴 × 퐵 × 퐶 = {푎, 푏, 푐|푎∈퐴,푏∈퐵,푐∈퐶} Cartesian product of sets 퐴, 퐵, 퐶; 퐸 universal set; 푃(퐸) = {푋|푋 ⊆ 퐸} set of parts (subsets) of set 퐸; 푑푒푓 퐴 = 퐵 ⇔ (∀) 푥 ∈ 퐸(푥 ∈ 퐴 ⇔ 푥 ∈ 퐵) equality of sets 퐴 and 퐵; 푑푒푓 퐴 ⊆ 퐵 ⇔ (∀) 푥 ∈ 퐸(푥 ∈ 퐴 ⇔ 푥 ∈ 퐵) 퐴 is included in 퐵; 퐴⋃퐵 = {푥 ∈ 퐸|푥 ∈ 퐴 ∨ 푥 ∈ 퐵} union of sets 퐴 and 퐵; 퐴⋂퐵 = {푥 ∈ 퐸|푥 ∈ 퐴 ∧ 푥 ∈ 퐵} intersection of sets 퐴 and 퐵; 퐴 ∖ 퐵 = {푥 ∈ 퐸|푥 ∈ 퐴 ∧ 푥 ∉ 퐵} difference between sets 퐴 and 퐵; 퐴∆퐵 = (퐴 ∖ 퐵) ∪ (퐵 ∖ 퐴) symmetrical difference; ∁휀(퐴) = 퐴̅ = 퐸 ∖ 퐴 complement of set 퐴 relative to 퐸; 훼 ⊆ 퐴 × 퐵 relation 훼 defined on sets 퐴 and 퐵; 푓: 퐴 → 퐵 function (application) defined on 퐴 with values in 퐵; 퐷(푓) domain of definition of function 푓: 퐸(푓) domain of values of function 푓. 8 Algebraic problems and exercises for high school 1. Sets. Operations with sets 1.1. Definitions and notations It is difficult to give an account of the axiomatic theory of sets at an elementary level, which is why, intuitively, we shall define a set as a collection of objects, named elements or points of the set. A set is considered defined if its elements are given or if a property shared by all of its elements is given, a property that distinguishes them from the elements of another set. Henceforth, we shall assign capital letters to designate sets: 퐴,퐵,퐶,…,푋,푌,푍, and small letters for elements in sets: 푎,푏,푐,…,푥,푦,푧 etc. If 푎 is an element of the set 퐴, we will write 푎 ∈ 퐴 and we will read "푎 belongs to 퐴" or "푎 is an element of 퐴". To express that 푎 is not an element of the set 퐴, we will write 푎 ∉ 퐴 and we will read "푎 does not belong to 퐴". Among sets, we allow the existence of a particular set, noted as ∅, called the empty set and containing no element. The set that contains a sole element will be noted with {푎}. More generally, the set that doesn’t contain other elements except the elements 푎1, 푎2, … , 푎푛 will be noted by {푎1, 푎2, … , 푎푛}. If 퐴 is a set, and all of its elements have the quality 푃, then we will write 퐴 = {푥|푥 verifies 푃} or 퐴 = {푥|푃(푥)} and we will read: "퐴 consists of only those elements that display the property 푃 (for which the predicate 푃( 푥) is true)." We shall use the following notations: ℕ = {0,1,2,3,… } – the natural numbers set; ∗ ℕ = {0,1,2,3,… } – the natural numbers set without zero; ℤ = {…,−2,−1,0,1,2,…} – the integer numbers set; ℤ = {±1,±2,±3…} – the integer numbers set without zero; 푚 ∗ ℚ = { | 푚, 푛 ∈ ℤ, 푛 ∈ ℕ } – the rational numbers set; 푛 9 Ion Goian ■ Raisa Grigor ■ Vasile Marin ■ Florentin Smarandache ℚ∗ = the rational numbers set without zero; ℝ = the real numbers set; ℝ∗ = the real numbers set without zero; ∗ ℝ+ = {푥 ∈ ℝ|푥 ≥ 0}; ℝ+ = {푥 ∈ ℝ|푥 > 0}; ℂ = {푎 + 푏|푎, 푏 ∈ 푅, 2 = −1} = the complex numbers set; ℂ∗ = the complex numbers set without zero; 푚 ∈ {1, 2, … , 푛} ⟺ 푚 = 1,̅̅̅̅ 푛̅; 퐷(푎) = {푐 ∈ ℤ∗|푎 ⋮ 푐} = the set of all integer divisors of number 푎 ∈ ℤ; 푛(퐴) = |퐴| = the number of the elements of finite set A. Note. We will consider that the reader is accustomed to the symbols of Logic: conjunction ∧ (…and…), disjunction ∨ (…or…), implication ⟹, existential quantification (∃) and universal quantification (∀). Let 퐴 and 퐵 be two sets. If all the elements of the set 퐴 are also elements of the set 퐵, we then say that 푨 is included in 푩, or that 푨 is a part of 푩, or that 푨 is a subset of the set 푩 and we write 퐴 ⊆ 퐵. So 퐴 ⊆ 퐵 ⇔ (∀)푥(푥 ∈ 퐴 ⟹ 푥 ∈ 퐵). The properties of inclusion a. (∀) 퐴, 퐴 ⊆ 퐴 (reflexivity); b. (퐴 ⊆ 퐵 ∧ 퐵 ⊆ 퐶) ⟹ 퐴 ⊆ 퐶 (transitivity); c. (∀)퐴, ∅ ⊆ 퐴. If 퐴 is not part of the set 퐵, then we write 퐴 ⊈ 퐵, that is, 퐴 ⊈ 퐵 ⇔ (∃)푥(푥 ∈ 퐴 ∧ 푥 ∉ 퐵). We will say that the set 퐴 is equal to the set 퐵, in short 퐴 = 퐵, if they have exactly the same elements, that is 퐴 = 퐵 ⟺ (퐴 ⊆ 퐵 ∧ 퐵 ⊆ 퐴). The properties of equality Irrespective of what the sets 퐴, 퐵 and 퐶 may be, we have: a. 퐴 = 퐴 (reflexivity); b. (퐴 = 퐵) ⟹ (퐵 = 퐴) (symmetry); 10 Algebraic problems and exercises for high school c. (퐴=퐵∧퐵 = 퐶)⟹(퐴=퐶) (transitivity). With 푃(퐴) we will note the set of all parts of set 퐀, in short 푋 ∈ 푃(퐴) ⟺ 푋 ⊆ 퐴. Obviously, ∅, 퐴 ∈ 푃(퐴). The universal set, the set that contains all the sets examined further, which has the containing elements’ nature one and the same, will be denoted by 퐸. Operations with sets Let 퐴 and 퐵 be two sets, 퐴, 퐵 ∈ 푃(퐸). 1. Intersection. 퐴 ∩ 퐵 = {푥 ∈ 퐸|푥 ∈ 퐴 ∧ 푥 ∈ 퐵}, i.e. 푥 ∈ 퐴 ∩ 퐵 ⟺ (푥 ∈ 퐴 ∧ 푥 ∈ 퐵), (1) 푥 ∉ 퐴 ∩ 퐵 ⟺ (푥 ∉ 퐴 ∧ 푥 ∉ 퐵), (1’) 2. Union. 퐴 ∪ 퐵 = {푥 ∈ 퐸|푥 ∈ 퐴 ∨ 푥 ∈ 퐵}, i.e. 푥 ∈ 퐴 ∪ 퐵 ⟺ (푥 ∈ 퐴 ∨ 푥 ∈ 퐵), (2) 푥 ∉ 퐴 ∪ 퐵 ⟺ (푥 ∉ 퐴 ∨ 푥 ∉ 퐵), (2’) 3. Difference. 퐴 ∖ 퐵 = {푥 ∈ 퐸|푥 ∈ 퐴 ∧ 푥 ∉ 퐵}, i.e. 푥 ∈ 퐴 ∖ 퐵 ⟺ (푥 ∈ 퐴 ∧ 푥 ∉ 퐵), (3) 푥 ∉ 퐴 ∖ 퐵 ⟺ (푥 ∉ 퐴 ∨ 푥 ∈ 퐵), (3’) 4. The complement of a set. Let 퐴 ∈ 푃(퐸). The difference 퐸 ∖ 퐴 is a subset of 퐸, denoted 퐶퐸(퐴) and called “the complement of 퐴 relative to 퐸”, that is 퐶퐸(퐴) = 퐸 ∖ 퐴 = {푥 ∈ 퐸|푥 ∉ 퐴}. In other words, 푥 ∈ 퐶퐸(퐴) ⟺ 푥 ∉ 퐴, (4) 푥 ∉ 퐶퐸(퐴) ⟺ 푥 ∈ 퐴. (4’) 11 Ion Goian ■ Raisa Grigor ■ Vasile Marin ■ Florentin Smarandache Properties of operations with sets 퐴∩퐴=퐴,퐴∪퐴=퐴 (idempotent laws) 퐴∩퐵=퐵∩퐴,퐴∪퐵=퐵∪퐴 (commutative laws) (퐴 ∩ 퐵) ∩ 퐶 = 퐴 ∩ (퐵 ∩ 퐶); (퐴 ∪ 퐵) ∪ 퐶 = 퐴 ∪ (퐵 ∪ 퐶) (associativity laws) 퐴 ∪ (퐵 ∩ 퐶) = (퐴 ∪ 퐵) ∩ (퐴 ∪ 퐶); 퐴 ∩ (퐵 ∪ 퐶) = (퐴 ∩ 퐵) ∪ (퐴 ∩ 퐶) (distributive laws) 퐴 ∪ (퐴 ∩ 퐵) = 퐴; 퐴 ∩ (퐴 ∪ 퐵) = 퐴 (absorption laws) 퐶퐸(퐴 ∪ 퐵) = 퐶퐸(퐴) ∩ 퐶퐸(퐵); 퐶퐸(퐴 ∩ 퐵) = 퐶퐸(퐴) ∪ 퐶퐸(퐵) (Morgan’s laws) Two "privileged" sets of 퐸 are ∅ and 퐸.