Homework

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.

- Homework 1.
- Homework 2.
- Homework 3.
- Homework 4.
- Homework 5.
- Homework 6.
- Homework 7.
- Homework 8.
- Homework 9.
- Homework 10.
- Homework 11.
- There will be no written homework due the last week of class.

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.

- 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*ax*^{6}+*bx*^{4}+*bx*^{2}+*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}=*A*union^{c}*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*(*n*^{2}- 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*I*= [_{n}*n*,*n*+1), an interval of real numbers. Prove that {*I*:_{n}*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*=3*y***R**to**R**, but does**not**define a function from**N**to**N** - Due Tuesday, November 28: Prove that the function
*f*(*x*)=*x*^{2}+5*x*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*:**P**→**P**by*f*(*x*) = (2*x*+ 3)/*x*. Prove that*f*is one-to-one, but not onto. [This is the last peer assessment homework for the semester.]