Advanced High-School Mathematics

SECTION 4.1 Basics of Set Theory 191 I’m sure that you’re reasonably comfortable with these notions. Two other important constru ...

192 CHAPTER 4 Abstract Algebra Actually, though, the De Morgan Laws are hardly surprising. IfA represents “it will rain on Monda ...

SECTION 4.1 Basics of Set Theory 193 Proof. As you might expect the above can be easily demonstrated through Venn diagrams (see ...

194 CHAPTER 4 Abstract Algebra Show that if A, B, and C ⊆ U, and if A, B, and C are finite subsets, then |A∪B∪C|=|A|+|B|+|C|−| ...

SECTION 4.1 Basics of Set Theory 195 A∩(B+C) = (A∩B) + (A∩C), whereA, B, C⊆U A+ (B∩C) = (A+B)∩(A+C), whereA, B, C ⊆U. Letpbe a ...

196 CHAPTER 4 Abstract Algebra see how the productS×S of two circles could be identified with the torus(the surface of a doughnu ...

SECTION 4.1 Basics of Set Theory 197 4.1.4 Mappings between sets LetAandB be sets. Amappingfrom Ato B is simply a function fromA ...

198 CHAPTER 4 Abstract Algebra If f :A→B is a mapping we callAthedomainof f and callB thecodomainoff. Therangeoff is the subset{ ...

SECTION 4.1 Basics of Set Theory 199 Suppose thatf :R → R is a cubic function such thatf′(x) 6 = 0 for all x∈ R. Give an intuit ...

200 CHAPTER 4 Abstract Algebra 4.1.5 Relations and equivalence relations LetSbe a set. ArelationRonSis simply a subset ofS×S. No ...

SECTION 4.1 Basics of Set Theory 201            1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 ...

202 CHAPTER 4 Abstract Algebra relation is symmetric. How about the transitivity? This involves a more work, but a bit of though ...

SECTION 4.1 Basics of Set Theory 203 [4] ={...,− 10 ,− 3 , 4 , 11 , 14 , ...} [5] ={...,− 9 ,− 2 , 5 , 7 , 12 , ...} [6] ={...,− ...

204 CHAPTER 4 Abstract Algebra Exercises LetS={ 1 , 2 , 3 , 4 }. How many relations are there onS? Let m∈Z+ and show that “≡ (m ...

SECTION 4.1 Basics of Set Theory 205 We saw on page 141 that the complete graphK 5 cannot be planar, i.e., cannot be drawn in t ...

206 CHAPTER 4 Abstract Algebra Form an equivalence relationRonS^2 by declaring any point on the sphere to be equivalent with its ...

SECTION 4.2 Basics of Group Theory 207 I have asked this question many times and to many people—some mathematicians—and often, i ...

208 CHAPTER 4 Abstract Algebra Aut(G). The most important facts related to graph automorphisms is the following: Proposition.The ...

SECTION 4.2 Basics of Group Theory 209 (i) Give an automorphismσ which takes vertex 1 to vertex 2 by completing the following: σ ...

210 CHAPTER 4 Abstract Algebra (i) Give an automorphismσwhich takes vertex 1 to vertex 2 by completing the following: σ:    ...