Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) 1. Suppose n e Z. If n² is even, then n is even. 2. Suppose ne Z. If n2 is odd, then n is odd. 3. Suppose a, b e Z. If a²(b²-26) is odd, then a and b are odd. 4. Suppose a,b,c e Z. If a does not divide bc, then a does not divide b. 5. Suppose x E R. If x² +5x<0 then x < 0. 6. Suppose x E R. If x³ - 7. Suppose a, b e Z. If both ab and a +b are even, then both a and b are even. 8. Suppose x E R. If x5 - 4x4 +3x³ -x² + 3x-4≥0, then x ≥ 0. -x>0 then x>-1.
Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) 1. Suppose n e Z. If n² is even, then n is even. 2. Suppose ne Z. If n2 is odd, then n is odd. 3. Suppose a, b e Z. If a²(b²-26) is odd, then a and b are odd. 4. Suppose a,b,c e Z. If a does not divide bc, then a does not divide b. 5. Suppose x E R. If x² +5x<0 then x < 0. 6. Suppose x E R. If x³ - 7. Suppose a, b e Z. If both ab and a +b are even, then both a and b are even. 8. Suppose x E R. If x5 - 4x4 +3x³ -x² + 3x-4≥0, then x ≥ 0. -x>0 then x>-1.
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:Exercises for Chapter 5
A. Prove the following statements with contrapositive proof. (In each case, think
about how a direct proof would work. In most cases contrapositive is easier.)
1. Suppose n e Z. If n² is even, then n is even.
2. Suppose ne Z. If n² is odd, then n is odd.
3. Suppose a, b e Z. If a²(b² - 2b) is odd, then a and b are odd.
4. Suppose a,b,c e Z. If a does not divide bc, then a does not divide b.
5. Suppose x E R. If x² + 5x<0 then x < 0.
6. Suppose x E R. If x³ -x>0 then x>-1.
7. Suppose a, b e Z. If both ab and a + b are even, then both a and b are even.
8. Suppose x E R. If x5 - 4x4 +3x³ - x² + 3x-4 ≥0, then x ≥ 0.
9. Suppose ne Z. If 3|n², then 3 n.
10. Suppose x, y, ze Z and x 0. If xyz, then xy and xtz.
11. Suppose x,y e Z. If x²(y + 3) is even, then x is even or y is odd.
12. Suppose a € Z. If a² is not divisible by 4, then a is odd.
13. Suppose x E R. If x5 +7x³ +5x2x4+x² +8, then x ≥ 0.
Expert Solution
Step 1
Contrapositive Form:
If we have the proposition as i.e., , then the contrapositive form of the given proposition is given by where is the negation symbol.
The contrapositive proof will be of the form:
Suppose
Therefore, .
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