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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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