Pre-HW, Practice HW and HW
Pre-Assignment 1.
(a) Show that sqrt{3} is irrational.
(b) Solve this. Assume each of the next three statements is true.
(i) John is smart. (ii) John or Mary is 10 years old.
(iii) If Mary is 10, then John is not smart.
Which is true? (1) Mary is 10. (2) John is 10. (3) Either John or Mary is not 10.
(c) Think about whether the sum of three consecutive numbers is divisible
by 3.
More Practice.
Page 70: 22(a), 24-25.
Page 71: D1.
Page 78: 3, 5, 7, 10, D2(Page 80).
Assignment 1.
Page 12: 1, 2, 5, 6 (More to come?)
Assignment 2.
As mentioned in class you have to prove the n|k^2 result.
Page 35-38: 1(b,c,d), 4, 9, 10, 11, 16, D3
Assignment 3. (As mentioned in class this is due on Thursday)
Page 57: 5, 6, 7(g,h), 9 (also if C is all the multiples of 5, then describe B intersect C), 13, 14, 19, 20, D1
Assignment 4.
Page 79: 17 (see also 18)
Page 157: 10
Page 169: 1c, 13a, 40
Assume that for any two positive numbers, a and b, (the square root of a times b= ) (a*b)^{1/2} (is less than or equal to) \le 1/2 *(a+b). Prove that for any n\ge 1,
if a_1, a_2, ..., a_n > 0 then (a_1 + a_2 + ... + a_n)/n \ge (a_1*a_2*...*a_n)^{1/n} (that is, the average is greater than or equal to the nth root of the product).
Assignment 5.
Due on Thursday, April 14: #1(below), 118/4,5 , 180/3,7,8,10
1. Let f be a function from X to Y. Define F from the power set of Y to the
power set of X by: for a subset, B, of Y, let F(B)=f^{-1}(B). That is, the inverse image of B under the map f. (a) Show that F is 1-1 iff f is onto and
(b) F is onto iff f is 1-1. (Check out this hint.)
2. Page 118: 2 -- 5, 7, 8, 9, 11, 13, D2
3. Page 180: 1b, 2 -- 8, 10 -- 17, D3, D4
Here are some more practice problems.
Page 197: 28, 29, 30, D1b
Some Practice.
Chapter 1.
Page 12: 8, 9, 10, D1, D5, D8
Page 26: 2, 4-7, 9, 11, 14, D1, D2, D4