Bridge to Abstract Mathematics: Mathematical Proof and Structures

5.2 CONCLUSIONS INVOLVING V AND +, BUT NOT 3 163 Solution You may recall this result as one that we assumed and used in the solu ...

164 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 As we did after Example 7, let us rewrite the preceding proof with ex- plana ...

5.2 CONCLUSIONS INVOLVING V AND -+, BUT NOT 3 165 only if there exist real numbers x and y such that (x, y) E C, but (x, -Y) 4 c ...

166 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 and B $ A. To prove the latter, we must prove (3x)[(x E B) A (x $ A)]. This ...

5.2 CONCLUSIONS INVOLVING V AND +, BUT NOT 3 lm definition of C, we know that (-x, f(-x)) E C. Since C is the graph of a functio ...

168 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 Recall (Definition 5, Article 1.1) that the power set 9(A) of a set A is th ...

5.2 CONCLUSIONS INVOLVING V AND +. BUT NOT 3 169 (b) Let C, be the curve in the xy plane described parametrically by x = cosh t ...

170 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 and x, be real numbers such that x, < x,. Then M = M(x, - x,)/(x, - x,) = ...

5.3 PROOF BY SPECIALIZATION AND DIVISION INTO CASES 171 to write out the proofs. Don't, however, become discouraged if you can't ...

172 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 Note that the proof of Example 3 employed a straightforward substi- tution, ...

5.3 PROOF BY SPECIALIZATION AND DIVISION INTO CASES 173 situations), may work to our advantage. Other situations will be demon- ...

174 METHODS OF MATHEMATICAL PROOF, PART 1 Chapter 5 used the other. Finally, notice the relationship between the result proved i ...

5.3 PROOF BY SPECIALIZATION AND DIVISION INTO CASES 175 (c) Prove or disprove the converse of the theorem proved in Example 8, A ...

176 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 (a) Prove that if a subset C of R x R is symmetric with respect to both the ...

5.3 PROOF BY SPECIALIZATION AND DIVISION INTO CASES 177 Let A = (aij), , , and B = (bij), , , be square matrices. Recall Exerci ...

178 METHODS OF MATHEMATICAL PROOF, PART 1 Chapter 5 5.4 Proof by Mathematical Induction Proof by mathematical induction is a spe ...

5.4 PROOF BY MATHEMATICAL INDUCTION 179 To a very large extent, theorems whose statement involves the phrase "for all positive i ...

160 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 prove this, it is sufficient, by Theorem 1, to prove (i) 1 E S and (ii) for ...

5.4 PROOF BY MATHEMATICAL INDUCTION 181 Students sometimes complain that the actual induction proof seems like an afterthought, ...

182 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 Solution Let S be the truth set ofp(n): x=, (k/2k) = 2 - [(n + 2)/2"]. To pr ...