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Abstract Algebra - Preliminaries

Abstract Algebra - Preliminaries

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- 1. Chapter 1- Preliminaries 1.1 History of Algebra The word “algebra”-al jebr (in Arabic) • was first used by Mohammed Al-Khwarizmi -muslim Math. • ninth century, when taught mathematics in Baghdad. • means “reunion”,decribes his method for collecting the terms of an equations in order to solve it. • Omar Khayyam, another Mathematician, defined it as the science of solving equations. Elementary Algebra (Classical age of Algebra) • its central theme is clearly identified as the solving of eqs. - method of solving linear, qudratic, cubic, quartic equations. - 1824, Niels Abel – there does not exits any formula for 1 equations degree 5 or greater.
- 2. Modern Age • new varieties of algebra arose connection with the application in math to practical problems. - Matrix Algebra - Bolean Algebra - Algebra of vectors and tensors - ~200 different kinds of algebra. • the awareness grew - algebra can no longer be conceived merely as the science of solving equations. - It had to be viewed as much more broadly as a branch of mathematics. revealing general principles which apply equally to all known and all possible algebras: * What is it that all algebras have in common? * What trait do they share which lets us refer to all of them as algebras? Algebraic Structure • Abstract Algebra (Modern Algebra) -more adv. course 2 The study algebraic structures.
- 3. 1.2 Logic and Proof Undefined Terms - Understand these terms and feel comfortable using them to define new terms. Important Terms • Statement or Proposition - Declarative sentence that is either true or false, but not both. • Postulates - Statements that are assumed to be true. • Definition - A precise meaning to a mathematical term. 3
- 4. • Theorem - A major landmark in the mathematical theory. - Postulates and definitions are used to prove theorems. - Once a theorem is proved to be true, it can be used. • Lemma - A result that is needed to prove a theorem. • Corollary - A result that follows immediately from a theorem. • Example - Is not a general result but is a particular case. • Proof - Mathematical argument intended to convince us that a result is correct. 4
- 5. Conjunction, Disjunction and Negation Definition: Let P and Q be statements. i) The statement P AND Q, P ∧ Q, is called the conjunction of P and Q. ii) The statement P OR Q, P ∨ Q, is called the disjunction of P and Q. iii) The negation of P is denoted by NOT P or ~ P Conditional and Biconditional Statement Conditional statement: “If P then Q”, P ⇒ Q. Biconditional statement: “P if and only if Q”, P ⇔ Q. 5
- 6. Quantifiers Consider a statement P(x) : x >5 - Statement P(x) is depending on the variable x. - Adding quantifiers can convert statement P(x) into a statement that is either true or false. • Universal quantifier (∀) P(x) is true for all values of x, denoted by ∀x, P ( x) or For all x, P(x). For every x, P(x). For each x, P(x). P(x), for all x. 6
- 7. • Existential quantifier (∃) There exist an x for which P(x) is true or For some x, P(x). P(x), for some x. : ∃x, P ( x) Example 1. ∀x ∈ R, x −1 = ( x −1)( x + x + 1) 3 2 - True or false statement? Why? 2. ∀x ∈ R, x + x − 6 = 0 2 - True or False statement? Why? 3. ∃x ∈ R, x + x − 6 = 0 2 - True or False statement? Why? 7
- 8. Proofs - Many mathematical theorems can be expressed symbolically in the form of P ⇒Q Assumption Or hypothesis Conclusion may consists of one or more statements. - The theorem says that if the assumption is true than the conclusion is true. - How do you go about thinking up ways to prove a theorem? • Understand the definitions • Try examples • Try standard proof methods 8
- 9. Methods of Proof ( P ⇒ Q) 1. Direct Method • find a series of statements P1,P2,…,Pn • verify that each of the implications below is true P →P , P →P2 , P2 →P3 .....Pn −1 →Pn and Pn → Q 1 1 Example An integer n is defined to be even if n = 2m for some integer m. Show that the sum of two even integers is even. Proof 9
- 10. 2. Contrapositive Method • may prove ¬Q → ¬P Example Proposition: If x is a real number such that x + 7 x < 9, then x < 1.1 3 2 Proof 10
- 11. 3. Proof by Contradiction • assume that P is true and not Q is true (Q is false) • will end up with a false statement S • Conclude that not Q must be false, i.e., Q is true Example Proposition: If x is an integer and x2 is even then x is an even integer. Proof 11
- 12. 4. Proof by Induction • assume that for each positive integer n, a statement P(n) is given. If 1. P(1) is a true statement; and 2. Whenever P(k) is a true statement, then P(k+1) is also true, • then P(n) is a true statement for every n in positive integer. Example Prove: 1 1 1 1 n + + + ... + = 1• 3 3 • 5 5 • 7 (2n − 1)(2n + 1) 2n + 1 Proof 12
- 13. 5. Proof by Counterexamples • Sometimes a conjectured result in mathematics is not true. • Would not be able to prove it. • Could try to disprove it. • The conjecture in the form of ∀x, P ( x ) • Take the negation: NOT (∀x, P ( x )) Equivalent to: ∃x, NOT P ( x) • Hence to disprove the statement ∀x, P ( x) need only to find one value, say c, such that P(c) is false. • The value c is called a counterexample to the conjecture. 13
- 14. Example Let x be a real number. Disprove the statement If x2 >9 then x >3. Solution Remark • To disprove the conjecture in the form of ∃x, P ( x) cannot use counter example!!! Its negation is equivalently in the form of ∀x, NOT Need to show that P(x) is false for all values of x. • To prove P ( x) P ⇔ Q : Prove P ⇒Q and Q ⇒P 14

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