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H 26. For all integers a, b, and c, if ab c then a/c and bc.

H 26. For all integers a, b, and c, if ab c then a/c and bc.

Advanced Engineering Mathematics
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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a. Write the statement formally using a quantifier and a variable. b. Determine whether the statement is true or false and justify your answer. 19. Show that the following statement is false: For all integers a and b, if 3 (a+b) then 3 (a - b). For each statement in 20-32, determine whether the statement is true or false. Prove the statement directly from the definitions if it is true, and give a counterex- ample if it is false. H 20. The sum of any three consecutive integers is divis- ible by 3. 21. The product of any two even integers is a multiple of 4. H 22. A necessary condition for an integer to be divisible by 6 is that it be divisible by 2. 23. A sufficient condition for an integer to be divisible by 8 is that it be divisible by 16. 24. For all integers a, b, and c, if a b and a c then a (2b-3c). 25. For all integers a, b, and c, if a is a factor of c and b is a factor of c then ab is a factor of c. H 26. For all integers a, b, and c, if ab c then a c and b|c. H 27. For all integers a, b, and c, if a (b + c) then a b or a c. 28. For all integers a, b, and c, if a bc then a b or a c. 2162 29. For all integers a and 3
Transcribed Image Text:a. Write the statement formally using a quantifier and a variable. b. Determine whether the statement is true or false and justify your answer. 19. Show that the following statement is false: For all integers a and b, if 3 (a+b) then 3 (a - b). For each statement in 20-32, determine whether the statement is true or false. Prove the statement directly from the definitions if it is true, and give a counterex- ample if it is false. H 20. The sum of any three consecutive integers is divis- ible by 3. 21. The product of any two even integers is a multiple of 4. H 22. A necessary condition for an integer to be divisible by 6 is that it be divisible by 2. 23. A sufficient condition for an integer to be divisible by 8 is that it be divisible by 16. 24. For all integers a, b, and c, if a b and a c then a (2b-3c). 25. For all integers a, b, and c, if a is a factor of c and b is a factor of c then ab is a factor of c. H 26. For all integers a, b, and c, if ab c then a c and b|c. H 27. For all integers a, b, and c, if a (b + c) then a b or a c. 28. For all integers a, b, and c, if a bc then a b or a c. 2162 29. For all integers a and 3
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