Homework Week 5
Discrete Mathematics · 13 problems · solutions not included
Assigned: 9, 10, 11, 13, 19, 23, 28. Read proof of equivalence on page 92 and examples 13 and 14, then do 38 and 43.
Exact problem text for these assigned textbook exercises was not pasted in chat.
Assigned: 3, 5, 7, 9, 13, 15, 21, 25, 31. Read example 23, then do 41 and 42.
Exact problem text for these assigned textbook exercises was not pasted in chat.
Review Problem 1
Proposition: If \(x\) is rational and \(y\) is irrational, then \(x+y\) is irrational.
Review Problem 2
Proposition: The product of two rational numbers is rational.
Review Problem 3
Statement to disprove: The product of two irrational numbers is irrational.
Review Problem 4
Proposition: If \(x\) is irrational, then \(1/x\) is irrational.
Review Problem 5
Proposition: \(n\) is even if and only if \(7n+4\) is even.
Review Problem 6
Proposition: If \(n^3+5\) is odd, then \(n\) is even.
Review Problem 7
Proposition: \(\max(x,y)+\min(x,y)=x+y\).
Review Problem 8
Proposition: \(|x|+|y|\ge |x+y|\).
Review Problem 9
Proposition: There exists a pair of consecutive integers such that one is a perfect cube and the other is a perfect square.
Review Problem 10
Proposition: If \(n\) is odd, then there exists a unique integer \(k\) such that \(n=(k-2)+(k+3)\).
Review Problem 11
Proposition:
\[ \min(x,y)=\frac{x+y-|x-y|}{2},\qquad \max(x,y)=\frac{x+y+|x-y|}{2}. \]