If H is the subgroup in Z 12, then H2 = (a) 5(b) 14 (c) {3, 6, 9, 0}  {2}(d) {5, 8, 11, 2} (e) {5, 6, 9, 0}(f) {3, 2, 5}

Slides:

Advertisements

Similar presentations

Is it rational or irrational?

Advertisements

Lesson 10.4: Mathematical Induction

PROOF BY CONTRADICTION

If R = {(x,y)| y = 3x + 2}, then R -1 = (1) x = 3y + 2 (2) y = (x – 2)/3 (3) {(x,y)| y = 3x + 2} (4) {(x,y)| y = (x – 2)/3} (5) {(x,y)| y – 2 = 3x} (6)

……+(4n-3) = n(2n-1) P 1 = 1(2(1)-1)=1 check.

AB 11 22 33 44 55 66 77 88 99 10  20  19  18  17  16  15  14  13  12  11  21  22  23  24  25  26  27  28.

Chapter Primes and Greatest Common Divisors ‒Primes ‒Greatest common divisors and least common multiples 1.

A Quick Look at Quantified Statements. Why are Quantified Statements Important? The logical structure of quantified statements provides a basis for the.

A b c d e f g h j i k Graph G is shown. Which of the following is a trail but not a path? 1). acdefhg 2). hfdcbdfe 3). ghfdcbde 4). acdegik 5). None of.

Assume f: A  B and g: B  C are functions. Which are defined? (1) f  g (2) g  f(3) both(4) neither.

How do we start this proof? (a) Assume A n is a subgroup of S n. (b)  (c) Assume o(S n ) = n! (d) Nonempty:

What is the first line of the proof? a). Assume G has an Eulerian circuit. b). Assume every vertex has even degree. c). Let v be any vertex in G. d). Let.

What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.

Can Dijkstra’s Algorithm be modified in an obvious way to give the longest path in a graph? a). Yes b). No c). I have absolutely no idea.

Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.

How do we start this proof? (a) Let x  gHg -1. (b)  (c) Assume a, b  H (d) Show Nonempty:

Find all subgroups of the Klein 4- Group. How many are there?

What is the best way to start? 1.Plug in n = 1. 2.Factor 6n 2 + 5n Let n be an integer. 4.Let n be an odd integer. 5.Let 6n 2 + 5n + 4 be an odd.

1 More Applications of the Pumping Lemma. 2 The Pumping Lemma: Given a infinite regular language there exists an integer for any string with length we.

How many non-isomorphic tournaments with 10 vertices are there? a). 5 b). 10 c). 362,880 d). Over nine million.

Methods of Proof Leo Cheung. A Quick Review Direct proof Proof by contrapositive Proof by contradiction Proof by induction.

What is entry A in the matrix multiplication: ( a) 1 (b) -2(c) 5 (d) 11(e) 13 (f) 0.

Equivalence Relations: Selected Exercises

Partial Orderings: Selected Exercises

Chapter 10 Sequences, Induction, and Probability Copyright © 2014, 2010, 2007 Pearson Education, Inc Mathematical Induction.

Reflexive -- First sentence of proof is: (1) Let x  Z (2) Let (x,x)  R. (3) Let (x,x)  I (4) Let x  R.

What is the next line of the proof? a). Assume the theorem holds for all graphs with k edges. b). Let G be a graph with k edges. c). Assume the theorem.

Find [G : H] if G = Z 18 and H =. (a) 1.5(b) 3 (c) 6(d) 9 (e) 12(f) 18 (g) None of the above.

1 Strong Mathematical Induction. Principle of Strong Mathematical Induction Let P(n) be a predicate defined for integers n; a and b be fixed integers.

App IV. Supplements to the Theory of Symmetric Groups Lemma IV.1: x p = p x p –1 Let { h } & { v } be horizontal & vertical permutations of Young tableaux.

Finite Groups & Subgroups. Order of a group Definition: The number of elements of a group (finite or infinite) is called its order. Notation: We will.

Mathematical Induction. F(1) = 1; F(n+1) = F(n) + (2n+1) for n≥ F(n) n F(n) =n 2 for all n ≥ 1 Prove it!

9.4 Mathematical Induction

Great Theoretical Ideas in Computer Science for Some.

Chapter 9 Functions Let A and B be nonempty sets. A function f from A to B, written f: A  B is a relation from A to B with the property that every element.

I.3 Introduction to the theory of convex conjugated function.

Chapter 5: Permutation Groups  Definitions and Notations  Cycle Notation  Properties of Permutations.

Section 3.3: Mathematical Induction Mathematical induction is a proof technique that can be used to prove theorems of the form:  n  Z +,P(n) We have.

Math 344 Winter 07 Group Theory Part 2: Subgroups and Isomorphism

CS Lecture 14 Powerful Tools     !. Build your toolbox of abstract structures and concepts. Know the capacities and limits of each tool.

6.3.2 Cyclic groups §1.Order of an element §Definition 13: Let G be a group with an identity element e. We say that a is of order n if a n =e, and for.

SECTION 8 Groups of Permutations Definition A permutation of a set A is a function  ϕ : A  A that is both one to one and onto. If  and  are both permutations.

SECTION 10 Cosets and the Theorem of Lagrange Theorem Let H be a subgroup of G. Let the relation  L be defined on G by a  L b if and only if a -1 b 

LEARNING TARGET: STUDENTS WILL BE ABLE TO USE PROPERTIES OF MIDSEGMENTS AND WRITE COORDINATE PROOFS. FEBRUARY 12, Midsegment Theorem and Coordinate.

Lesson 1.6 Paragraph Proofs Objective: Write paragraph proofs.

EXAMPLE 3 Write an indirect proof Write an indirect proof that an odd number is not divisible by 4. GIVEN : x is an odd number. PROVE : x is not divisible.

Mathematical Induction. The Principle of Mathematical Induction Let S n be a statement involving the positive integer n. If 1.S 1 is true, and 2.the truth.

Section 2.2 Day 1. A) Algebraic Properties of Equality Let a, b, and c be real numbers: 1) Addition Property – If a = b, then a + c = b + c Use them 2)

Finite Groups and Subgroups, Terminology

Math 3121 Abstract Algebra I

Chapter 4: Cyclic Groups

B.Sc. III Year Mr. Shrimangale G.W.

What is the next line of the proof?

Proofs with >Out > >Out (1) A (2) A>S (3) S>U

Indirect Argument: Contradiction and Contraposition

An indirect proof uses a temporary assumption that

קבלת ההחלטה בנושא דרגת אי כושר

PROOF BY CONTRADICTION

Sec 21: Analysis of the Euler Method

Follow me for a walk through...

Direct Proof and Counterexample II

Follow me for a walk through...

Proof by Induction L.O. All pupils understand proof by induction

Rayat Shikshan Sanstha’s S.M.Joshi College, Hadapsar -28

Follow me for a walk through...

Multiplying integers 3(3) = (3) = (3) = (3) = 0 -3

%e0%a4%b8%e0%a5%8d%e0%a4%9f%e0%a5%8b%e0%a4%b0%e0%a5%80%e0%a4%9c/%e0% a4%86%e0%a4%b0%e0%a5%8d%e0%a4%ae%e0%a5%80-

%e0%a4%b8%e0%a5%8d%e0%a4%9f%e0%a5%8b%e0%a4%b0%e0%a5%80%e0%a4%9c/%e0% a4%86%e0%a4%b0%e0%a5%8d%e0%a4%ae%e0%a5%80-

Presentation transcript:

If H is the subgroup in Z 12, then H2 = (a) 5(b) 14 (c) {3, 6, 9, 0}  {2}(d) {5, 8, 11, 2} (e) {5, 6, 9, 0}(f) {3, 2, 5}

If H is the subgroup in Z 12, then H7 = (a) {3, 6, 9, 0}  {7}(b) {3, 7, 10} (c) {7, 10, 1, 4}(d) {7, 2, 9} (e) 10(f) 3  7

If H is the subgroup in S 3, then H(1,3) = (a) (1,2,1,3)(b) {(1,3,2), (1,3)} (c) {(1, 3, 2)}(d) {(1, 2, 3)} (e) {(1,2), (1,3)}(f) {(1,2,3), (1, 3)}

If H is the subgroup in S 3, then H(1,2) = (a) (1,2,1,2)(b) (2,4) (c) (1,2) 2 (d) {(1, 2)} (e) {(1,2), (2,1)}(f) H

If H is the subgroup in Z 4 x Z 6, then H(1,2) = (a) (2,1)(b) (3,4) (c) (2,2)  (1,2) (d) {(3,4), (1,0), (3,2), (1,4), (3,0), (1,2)} (e) {(1,2), (2,1), (3,4), (0,5), (1,1), (2,3)} (f) H

If H is the subgroup in Z 4 x Z 6, then H(2,0) = (a) (2,1)(b) (3,4) (c) (2,2)  (1,2) (d) {(3,4), (1,0), (3,2), (1,4), (3,0), (1,2)} (e) {(1,2), (2,1), (3,4), (0,5), (1,1), (2,0)} (f) H

How do we start this proof? (a) Assume A n is a subgroup of S n. (b)  (c) Assume a, b  H (d) Nonempty:

What is the next line of the proof? (a) Assume Ha = Hb (b) Assume ab -1  H (c) Assume a, b  H (d) Assume if ab -1  H then Ha = Hb.

What is the next line of the proof? (a) Assume Ha = Hb (b)  (c) Assume a, b  H (d) Assume if ab -1  H then Ha = Hb.

What is the next line of the proof? (a) Assume Ha = Hb (b) Assume ab -1  H. (c) Assume a, b  H (d) Assume if ab -1  H then Ha = Hb.

What is the next line of the proof? (a) Assume Ha = Hb (b) Then a = Hb (c) Then b -1 a  H(d) Let y  H (e) Let y  Ha(f) Let y = ab -1