# Math 3325 Principles of Math Homework

## Dr. Duval

For the week of November 21: On Tuesday we will discuss Section 4.2. Thursday is Thanksgiving, so no class!

For the week of November 28-30: On Tuesday we will continue to discuss Section 4.2. On Thursday we will continue to discuss Section 4.2, and start discussing Section 4.3.

For the week of December 5-7: On Tuesday, we will finish discussing Section 4.3, and begin discussing Section 4.4. On Thursday, we will continue to discuss Section 4.4.

### Discussion homework

1.2: 2ce, 6fg.
1.3: 4, 8acdl, 10bg.
1.4: 4bc, 5d, 6e, 7ch, 9c.
1.5: 3c, 4c, 5a, 6e, 7a.
1.6: 1b, 2d, 4c, 6gi.
1.7: 2, 3e.

2.1: 2, 5bdf, 6bf, 11a, 14d, 15d.
2.2: 2dhj, 7q, 8c, 9d, 15d.
2.3: 1j, 3b, 7b, 16f.
2.4: 4eh, 5f, 6b, 7b.
2.5: 3.

3.1: 2abcd, 4fh, 5a, 7ae, 8ef.
3.2: 1bdgf, 6b, 7.
3.3: 2abc, 3d.
3.3: 4d, 7a, 10a.

4.1: 1bcde, 3bf, 6b.
4.2: 1di, 2bg, 4b, 16b, 17bc.
4.3: 1bl, 2bl, discussed in class, Tuesday, December 5.
4.3: 6, 9b, discussed in class, postponed until Thursday, December 7.

### Homework for peer assessment

• Due Thursday, September 21: Prove that there is a natural number M such that if n is a positive integer, then 2/n < M.
• Due Tuesday, October 3: Prove that if a and b are integers, then 1 and -1 are not solutions to the equation ax6 + bx4 + bx2 + a = 1. [Note the changed statement, and delayed due date.]
• Due Thursday, October 5: Prove that {x element-of N: 12|x } is a subset of {x element-of N: 3|x }
• Due Tuesday, October 10: Prove that (A-B)c = Ac union B.
• Due Thursday, October 12: Prove that if A is a subset of B and C is a subset of D, then A x C is a subset of B x D.
• Due Tuesday, October 17: Prove that the union of the sets [0,n), as n goes from 1 to infinity, is the set of integers greater than or equal to 0. Here, [0,n) is defined to be the set {k element-of Z: 0 less-than-or-equal k < n}.
• Due postponed until Tuesday, October 24: Prove by induction that, for all natural numbers n, (1)(2) + (2)(3) + ... + (n)(n-1) = n(n2 - 1)/3.
• Due Thursday, October 26: Let R be a relation from A to B. Prove that Dom(R-1) = Rng(R). [Hint: Remember that domain and range are each sets. How do we show that sets are equal?]
• Due Tuesday, October 31: Prove that the relation T on the real numbers given by x T y iff x = y or x = -y is an equivalence relation.
• Due Thursday, November 2: Define the relation L on ordered pairs of real numbers given by (x,y) L (u,v) iff x + y = u + v. Prove that L is an equivalence relation, and describe the equivalence class of (2,3). Draw a picture in the (x,y)-plane showing this equivalence class.
• Due Tuesday, November 7: For each integer n, define In = [n, n+1), an interval of real numbers. Prove that {In: n element-of Z} is a partition of the real numbers.
• Due postponed until Tuesday, November 14: Prove that the relation T given by
x T y if x=3y
defines a function from R to R, but does not define a function from N to N
• Due Tuesday, November 28: Prove that the function f(x)=x2+5x is increasing on the interval [0,infinity) = {x element-of R: x greater-than-or-equal 0}. Do not use derivatives; just use the definition of increasing function.
• No peer assessment homework due Thursday, November 30.
• Due Tuesday, December 5: Let P denote the positive real numbers. Define a function f:PP by f(x) = (2x + 3)/x. Prove that f is one-to-one, but not onto. [This is the last peer assessment homework for the semester.]