E-LESSON-7

IDOL Institute of Distance and Online Learning ENHANCE YOUR QUALIFICATION, ADVANCE YOUR CAREER.

B.C.A 2 All right are reserved with CU-IDOL Mathematics Course Code: BCA114 Semester: First SLM UNITS : e-Lesson No.: 7 4 www.cuidol.in Unit-7 (BCA114)

Modern Algebra 4 33 OBJECTIVES INTRODUCTION Student will be able to : In this unit we are going to learn about the Define principle of Mathematical Induction. principle of mathematical induction. Illustrate the De Moivre’s theorem. Under this unit you will also understand the De Moivre’s theorem. Describe the term Quantifiers and its type. This Unit will also make us to understand the Quantifiers. www.cuidol.in Unit-7 (BCA114) INASllTITriUgThEt aOrFeDreISsTeArNveCdE AwNitDh OCNUL-IIDNOE LLEARNING

TOPICS TO BE COVERED 4 • Principle of mathematical Induction • De Moivre’s Theorem • Quantifiers www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

Principle of Mathematical Induction 5 Mathematical thinking is essential, as it helps in better understanding of problems thus letting us reach to solutions in a better way. Deductive reasoning and mathematical thinking are connected, as the former forms the basis of the latter. Deductive reasoning further is based on logic and understanding. See the example given underneath: 1. The German shepherd is a dog. 2. All dogs have an immaculate sense of hearing. 3. The German shepherd has an immaculate sense of hearing. Definition •Let Pn be a statement involving the natural numbers. 1. If P1 is time and 2. Truth of Pk  Truth of Pk+1 for each position integer k, then Pn is true for all integers. www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

Principle of Mathematical Induction 6 •Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), . . . . Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: •Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). • For any natural number n, n + 1 is greater than n. •For any natural number n, no natural number is between n and n + 1. •No natural number is less than zero. www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

De Moivre’s Theorem 7 •In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number x and integer n it holds that (cos(x)+isin(x)^n= cos(nx)+isin(nx) where i is the imaginary unit (i^2 = −1). •The formula is named after Abraham de Moivre, although he never stated it in his works. The expression cos(x) + i sin(x) is sometimes abbreviated to cis(x). •The formula is important because it connects complex numbers and trigonometry. •By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of cos(x) and sin(x). www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

Second Principle of Mathematical Induction 8 •Suppose 1. P(1) is true, and 2. If P(1), P(2), ..., P(k) are true, then P(k + 1) is also true. then P(n) is true for all positive integers. •The second principle of mathematical induction is usually stated and demonstrated for n0 being either 0 or 1. •This is often dependent upon whether the analysis of the fundamentals of mathematical logic are zero-based or one-based. www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

Quantifiers 9 •Universal Quantifier Some statements assert that a property is true for all values of a variable in a particular domain, known as universe of discourse. • Definition: “A proposition P(x) is true for all the values of x in a universe of discourse”, is the universal quantification of P(x). It is denoted by x P(x) . The following phrases also represent Universal Quantification: For all For every For each Example: Let W(x) be the proposition “the birds having a wing”. Since all birds have wings, the proposition is “for all birds”. Hence, x (W(x)) is true. www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

Quantifiers 10 • Existential Quantifier •Definition: The proposition, “there exists an element x in the universe of discourse such that P(x) is true” is known as existential quantifiers of P(x). The symbol x represents the existential quantifier. It represents each of the following phrases: For some x, Some x such that, There is at least one x such that, Example: Let P(x) be “x^2 < 10” and the universe of discourse be set of integers. There exists integers –3, –2, –1, 0, 1, 2, and 3 for which P(x) is true, i.e., P(–3), P(–2), P(–1), P(0), P(1), P(2) and P(3) are true. Therefore, x P(x) is true. www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

MULTIPLE CHOICE QUESTIONS 11 1. P be the proposition defined on integer n 1 such that ________. (a) P(1) is true (b) P(1) is false c) P(1) = 1 (d) None 2. In the principle of mathematical induction, which of the following steps is mandatory? a) induction hypothesis b) inductive reference c) induction set assumption d) minimal set representation 3. For m = 1, 2, …, 4m+2 is a multiple of ________ (a) 3 (b) 5 (c) 6 (d) 2 4. Let Q(x, y) denote “M + A = 0.” What is the truth value of the quantifications ∃A∀M Q(M, A) a) True b) False 5. Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. Use quantifiers to express , “Joy is loved by everyone.” a) ∀x L(x, Joy) b) ∀y L(Joy,y) c) ∃y∀x L(x, y) d) ∃x ¬L(Joy, x) Answers: 1.(a) 2. (a) 3.(d) 4. (b) 5.(a) www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

SUMMARY 12 Let us recapitulate the important concepts discussed in this session: •Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), . . . . Informal metaphors help to explain this technique. •In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number x and integer n it holds that (cos(x)+isin(x)^n= cos(nx)+isin(nx). •“A proposition P(x) is true for all the values of x in a universe of discourse”, is the universal quantification of P(x). It is denoted by x P(x) . •The proposition, “there exists an element x in the universe of discourse such that P(x) is true” is known as existential quantifiers of P(x). The symbol x represents the existential quantifier. www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

FREQUENTLY ASKED QUESTION 13 Q:1 Explain Principle of Mathematical Induction. Ans: Let Pn be a statement involving the natural numbers. 1. If P1 is time and 2. Truth of Pk  Truth of Pk+1 for each position integer k, then Pn is true for all integers. For more details refer subject SLM unit 7. Q2: Explain Universal Quantifier. Ans: “A proposition P(x) is true for all the values of x in a universe of discourse”, is the universal quantification of P(x). It is denoted by x P(x) . The following phrases also represent Universal Quantification: For all or For every or For each. For more details refer subject SLM unit 7. Q3. Explain Existential quantifiers. Ans. The proposition, “there exists an element x in the universe of discourse such that P(x) is true” is known as existential quantifiers of P(x). The symbol x represents the existential quantifier. It represents each of the following phrases: For some x or Some x such that. For more details refer subject SLM unit 7. www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

REFERENCES 14 CU-IDOL’s “Mathematics” SLM. Seymour Lipschutz, Marc Lars Lipson, “Discrete Mathematics”, Publisher: McGraw Hill Education (India) Private Limited.  J.K. Sharma, “Discrete Mathematics”, Publisher: MacMillan India Limited. K. Chandrasekhara Rao, “Discrete Mathematics”, Publisher: Narosa Publishing House. Dr. Abhilasha S. Magar, “Applied Mathematics – I”, Publisher: Himalaya Publishing House. Dr. Abhilasha S. Magar, “Business Mathematics”, Publisher: Himalaya Publishing House. Dr. Abhilasha S. Magar, “Quantitative Methods – II”. Publisher: Himalaya Publishing House. www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL

15 THANK YOU For queries Email: [email protected] www.cuidol.in Unit-7 (BCA114) All right are reserved with CU-IDOL