Bridge to Abstract Mathematics: Mathematical Proof and Structures

22 SETS Chapter 1 DEFINITION 7 Given sets A and B, we define the cartesian product A x B (read "A cross B," cartesian product is ...

1.2 OPERATIONS ON SETS 23 with reference to two perpendicular lines, the x axis and the y axis. I x J is the rectangle, open abo ...

24 SETS Chapter 1 Figure 1.2 Graphic interpretation of the mean value theorem: f ' = [f(b) - f (a)ll[rb - al. to the conjecture ...

1.2 OPERATIONS ON SETS 25 Figure 1.3 Venn diagram representation of intersection. Some of the other four operations described ea ...

26 SETS Chapter 1 -.. (A U B) n C.is the crosshatched region Figure 1.5 A more complicated Venn diagram representation. operatio ...

(c) (AuC)-(AnC) (d) AA C (e) B A U (f) A A (B A C) (g) (A A B) A C (h) C - (B - A) (i) (C - B) - A (j) (C - B) n (C - A) (k) C-( ...

I 28 SETS Chapter 1 Given U = (1, 2,3,4), A = (1, 3,4), B = (31, C = (1, 21, find all pairs of disjoint sets among the six sets ...

1.3 ALGEBRAIC PROPERTIES OF SETS 29 (b) Figure 1.6 (d) How many regions (i.e., nonoverlapping subsets of U having all of U as th ...

30 SETS Chapter 1 In that example U was not specified, nor was it totally clear from the context what U should be (R and Z would ...

1.3 ALGEBRAIC PROPERTIES OF SETS 31 COMMUTATIVITY AND ASSOCIATIVITY EXAMPLE2 LetU=(1,2,3 ,..., 9,lO),A={2,3,5,8),B=(1,2,5,6, 7, ...

32 SETS Chapter 1 CONJECTURE 2 Let X, Y, and Z be any sets. Then: (f) Xn(YnZ)=(XnY)nZ and Xu(YuZ)=(XuY)uZ (g) XnY=YnX and XuY=Yu ...

1.3 ALGEBRAIC PROPERTIES OF SETS 33 (A nB)UCisthe shaded region ~'n (B U C) is the crosshatched region Figure 1.8 Venn diagram s ...

34 SETS Chapter 1 To get further evidence, you should test this equation by using sets A, B, and C of Example 2 and also by draw ...

1.3 ALGEBRAIC PROPERTIES OF SETS 35 THEOREMS INVOLVING A CONDITIONAL As we progress in abstract mathematics, we find that the mo ...

1.3 ALGEBRAIC PROPERTIES OF SETS 37 standing of what the issues are, based on the following discussion, which is essentially an ...

38 SETS Chapter 1 (e) B u (A - B) = A (f) IfAnCcBnC,thenA~B *(g) IfX c A, then X u (A n B) = (X u A) n B (h) If A x B = A x C, t ...

1.4 THEOREMS OF SET THEORY 39 (b) A U B is the shaded region Figure 1.9 (c) x2 - 3x - 28 > 0 and Ix - $1 5 9 (d) Ix - $1 I an ...

40 SETS Chapter 1 "conjectures." You should already have conjectured, based on earlier ex- ercises, a large number of potential ...

1.4 THEOREMS OF SET THEORY 41 FACT 4 The following distributive laws can be proved to be theorems of set theory. For all sets XI ...

42 SETS Chapter 1 FACT 8 The following statements of equivalence, that is, involving "if and only if," can be proved to be theor ...