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1) Direct Demonstrations a. Show: 3x € ZZ, 2x + 1 = 7 b. Show: f(x) = C(R), f(x) has exactly 2 zeros c. Show: 3x Є R, e* = 2 2) Counter Examples a. Disprove: VzЄ Z, z z³ b. Disprove: Vf(x) = C(R), f²(x) > f(x) c. Disprove: VM, NE Mat2x3, MN E Mat2x3 3) Direct Proofs a. Show: Vx € Z,x even x³ even b. Show: Vx, y, x EZ, x even, y odd, z odd → xyz even 4) Contradiction Show: Vx, y € ZZ, z = x² + y² and x,y same parity → z is even (hint, there are 4 cases to check) 5) Contrapositive Show: Vz Є ZZ, z5 is odd→ z is odd

1) Direct Demonstrations a. Show: 3x € ZZ, 2x + 1 = 7 b. Show: f(x) = C(R), f(x) has exactly 2 zeros c. Show: 3x Є R, e* = 2 2) Counter Examples a. Disprove: VzЄ Z, z z³ b. Disprove: Vf(x) = C(R), f²(x) > f(x) c. Disprove: VM, NE Mat2x3, MN E Mat2x3 3) Direct Proofs a. Show: Vx € Z,x even x³ even b. Show: Vx, y, x EZ, x even, y odd, z odd → xyz even 4) Contradiction Show: Vx, y € ZZ, z = x² + y² and x,y same parity → z is even (hint, there are 4 cases to check) 5) Contrapositive Show: Vz Є ZZ, z5 is odd→ z is odd

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 52RE
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1) Direct Demonstrations a. Show: 3x € ZZ, 2x + 1 = 7 b. Show: f(x) = C(R), f(x) has exactly 2 zeros c. Show: 3x Є R, e* = 2 2) Counter Examples a. Disprove: VzЄ Z, z z³ b. Disprove: Vf(x) = C(R), f²(x) > f(x) c. Disprove: VM, NE Mat2x3, MN E Mat2x3 3) Direct Proofs a. Show: Vx € Z,x even x³ even b. Show: Vx, y, x EZ, x even, y odd, z odd → xyz even 4) Contradiction Show: Vx, y € ZZ, z = x² + y² and x,y same parity → z is even (hint, there are 4 cases to check) 5) Contrapositive Show: Vz Є ZZ, z5 is odd→ z is odd
Transcribed Image Text:1) Direct Demonstrations a. Show: 3x € ZZ, 2x + 1 = 7 b. Show: f(x) = C(R), f(x) has exactly 2 zeros c. Show: 3x Є R, e* = 2 2) Counter Examples a. Disprove: VzЄ Z, z z³ b. Disprove: Vf(x) = C(R), f²(x) > f(x) c. Disprove: VM, NE Mat2x3, MN E Mat2x3 3) Direct Proofs a. Show: Vx € Z,x even x³ even b. Show: Vx, y, x EZ, x even, y odd, z odd → xyz even 4) Contradiction Show: Vx, y € ZZ, z = x² + y² and x,y same parity → z is even (hint, there are 4 cases to check) 5) Contrapositive Show: Vz Є ZZ, z5 is odd→ z is odd
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