Math 366 Practice Problems for Exam 2Autumn 20071.Constructive Proof of an Existential Statement.(a) Prove that there is an even integernsuch thatn mod3 = 1.(b) Prove that there exists a rational numberqsuch that 9q2= 4.(c) Prove that there exist two real numbers whose product is less than their sum.(d) Prove that there exist two real numbers which are not equal to each other and whose productis equal to their sum.(e) Prove that there is an odd integernsuch thatn >1 andnhas the form 3k+ 1 for someintegerk.2. Direct Proof of a Universal Statement.(a) Prove that ifnis an integer which is divisible by 6 thennis divisible by 3.(b) Prove that for any integersa, b, c,andd, ifadividesbandcdividesdthena·cdividesb·d.(c) Prove that the product of two odd integers is odd.(d) Prove that ifnis an integer which is divisible by 5 then 3nis divisible by 15.(e) Prove that for any nonzero rational numbersaandbthere is a rational numberxsuch thatax+b= 0.
This is the end of the preview.
access the rest of the document.