HW # 1, due Monday, August 31

Read 1.1 - 1.3.

Do, But Don't Hand In

1.1 # 1, 2a-d, 3g,k,m, 4c,g
1.2 # 1a,d,f, 2a,d,f, 4a,b,d,g,h, 7e,g, 8a,c,e,f, 11

Hand In

1.1 # 3j,l, 4b,d,f, 7, 10a,d
1.2 # 6a,e, 9e,f (by truth table, or by Theorem 1.2), 14d,h,i



HW # 2, due Friday, September 11

Read 1.4-1.6.

Do, But Don't Hand In

1.3 # 1a,b,i,j, 2a,b,i,j, 4i,j, 6a,b
1.4 # 1 (excluded middle, contradiction, contrapositive,
         transitivity,  modus ponens), 5a,e, 10b,d,e,f

Hand In

1.3 # 1h,k, 2h,k, 4a,b,g,h, 6d,e, 9, 10c
       (explain your answers)
1.4 # 4b,c, 6e,h, 7, 8b
      (*) Let a and b be real numbers. Prove that
          if the square of a is less than or equal
	  to the square of b, then |a| is less than
	  or equal to |b|. 
     (**) Prove that the common (base-10) logarithm of 2 is irrational.\



HW # 3, due Monday, September 21

Do, But Don't Hand In

1.5 # 5b,c, 8a, 9d,e,f,g,i
1.6 # 11b,c

Hand In

1.5 # 1d,g,i, 2a,d, 5h, 7b,d, 8b,e,h
1.6 # 3a,b, 6, 9



HW # 4, due Monday, September 28

Do, But Don't Hand In

2.1 # 1a,b, 2a,b, 4a-f, 6a,d, 7c,e, 8a, 19a,c,f
2.2 # 1a,c,e,g,i, 8n, 9d,h, 14a, 15c, 17d,e,g,j

Hand In

2.1 # 9h,j,l (explain), 11, 12, 17
2.2 # 10b,e, 11d, 13b,c, 14f, 15b, 16f,g



HW # 5, due Monday, October 5

Do, But Don't Hand In

2.3 # 1j,k,l,m,n, 2j,k,l,m,n, 6, 11, 15, 18, 20f
2.4 # 2, 7, 8o, 9a,c, 15a,b,e

Hand In

2.3 # 4 (why must the family be nonempty?), 7,
      8a,b (prove your answers), 16,
     19b,c (in each family the sets should all be different)
2.4 # 4, 8b,e,j,m, 9d, 13



HW # 6, due Friday, October 9

Do, But Don't Hand In

2.5 # 5b, 15c

Hand In

2.5 # 6c, 7, 14c
    



TEST 1 --- Chapters 1 & 2

will be a take-home exam, given out in class on Friday,
October 9, and collected in class on Monday, October 12.

Rules

You may refer to YOUR OWN textbook and YOUR OWN notes and old quizzes.
You may NOT discuss the test problems or course material with ANY
other person, except me.  If you have a question about a problem,
you may ask me by e-mail, or phone (839-8424 between 9am and 9pm ---
be ready to leave a message in case I need to call you back later).




HW # 7, due Monday, October 19

Do, But Don't Hand In

2.6 # 17
3.1 # 1d, 3d, 4a,b, 6f,g,h, 7f,g,h, 8f, 9a,e,f,g,h
      10i, 11b,d,f

Hand In

2.6  (*) Give an algebraic proof of Theorem 2.25d.
    (**) Prove Theorem 2.25a by induction (using 2.25d).
3.1 # 3f, 12, 13



HW # 8, due Monday, October 26

Do, But Don't Hand In

3.2 # 1d,f,g,h, 2,d,f,g, 4d, 5a,d, 8, 9
3.3 # 2a,b, 3, 5a,c,d

Hand In

3.2 # 1j (prove your answer), 4b,g, 7
3.3 # 6, 9, 11, 12



HW # 9, due Monday, November 2

Do, But Don't Hand In

3.4 # 1, 2, 4, 5, 17, 19c, d

Hand In

3.4 # 6, 7, 8*, 15, 16**, 20b

*  Condition (i) should use 'strictly less than' rather
   than 'less than or equal to'.
** The least upper bound should be the union of all
   elements of B, and the greatest lower bound should
   be the intersection of all elements of B.  (Note
   that the elements of B are subsets of some unnamed
   set.)



HW # 10, due Monday, November 9

Do, But Don't Hand In

4.1 # 1f, h, i, j, 2f, g, 5c, d, e, f, 7c, d, 11, 16b
4.2 # 1f, j, 2f, j, 3b, f, 6, 8, 12b, c, e, 14e

Hand In

4.1 # 8c, d, f, 16d, (*)
4.2 # 9, (**)

 (*)  Let A and B be sets, let g : A -> B be a function,
 and let R be an equivalence relation on A.  Define a
 relation f from A/R to B by
 
   f = {(S,y) in A/R x B : there exists x in S with g(x)=y}
   
 Prove that f is a function <=> g is constant on equivalence
 classes.  (We say that "g is constant on equivalence classes"
 if (for all x,z in A) (xRz => g(x)=g(z)).)  Prove also that
 if f is a function, then for any x in A, f(x/R) = g(x).
   
 (**) Let I and J be intervals of the real line, let h
 be a real-valued function on I with range(h) contained
 in J, and let g be a real-valued function on J.  Let f
 be the composition of g with h.  Prove that f is increasing
 on I if h is increasing on I and g is increasing on J.



HW # 11, due Wednesday, November 18

Do, But Don't Hand In

4.3 # 1b, h, l, 2b, h, l, 3, 8, 15
4.4 # 5e, f, 6e, 9, 10b, d, 12, 14, 19

Hand In

4.3 # 4, 6(show by example that g need not be 1-1), 12
4.4 # 11, 13, 15, 17(give necessary and sufficient conditions,
                     with proofs)
                     


Test 2 Date Change

Test 2, covering 3.1-3.4 and 4.1-4.4 will be given in class
on Friday, November 20.



HW # 12, due Tuesday, December 1

Quiz:  Monday, November 30, Section 5.1.

Read:  5.1.

Do, But Don't Hand In

5.1 # 1, 9, 17

Hand In

5.1 # 2, 8, 14



HW # 13, due Monday, December 7

Read:  5.2.

Do, But Don't Hand In

  (*) Find an explicit equivalence between N x N x N and N
      (where N is the natural numbers).
 (**) Find an explicit equivalence between S and T, where
      S = { all real numbers greater than or equal to zero},
      T = { all real numbers greater than zero}.
(***) Let I = (0,1), the set of all real numbers strictly between
      0 and 1.  Define a function f : I x I -> I as follows:
      if (x,y) is in  I x I, take the decimal expansions of x
      and y in normalized form, and interleave them to get the
      decimal expansion of the number f(x,y).
      Prove that f is one-to-one.  Is f onto?  Prove your answer.
      
The last quiz will be given on Wednesday, December 2.
   

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