homework 4 (Due in class, on February 13th (Thursday)
(1) Calculate P(P(P(;)
(2) (Exercise 7.5) Using the given symbolization key, translate each Englishlanguage assertion into First order logic.
U : The set of all animals.
A : The set of all alligators.
Sample Midterm 2
The following is a sample set of questions for your preparation. Remember that
contents for this class are cumulative.
(1) Suppose A, B, and C are sets (and U is the universal set, as usual.)
(a) Prove that if A B and B C then A C.
(b) Pr
Sample Final
The following is a sample set of questions for your preparation. Remember that
contents for this class are cumulative.
(1) (a) Let f : A ! B. What does it mean to say that f is invertible?
(b) Show that if f : A ! B and g : B ! C are both inv
(1)
(2)
(3)
(4)
(5)
homework 7 (Due in class, on April 1st (Tuesday)
Show that if A B, then there exists an injective function f : A ! B.
Show that if A and B are non-empty sets so that A B, then there exists
a surjective function f : B ! A.
Show that if
MATH 2000 - Midterm
March 20, 2014
Last Name:
First Name:
Student ID:
Tutorial Time:
No external aid is allowed. Cell phones should be turned o before the start of this exam.
This exam is 60 minutes long.
There are 8 pages and 6 questions in this exam.
Ma
Sample Midterm
The following is a sample set of questions for your preparation. This will be too
long for an hour long midterm. (This should probably take you a bit more than
two hours to do.)
(1) For each assertion below, decide if it is a tautology, con
(1)
(2)
(3)
(4)
homework 6 (Due in class, on March 13th (Thursday)
Each formula below denes a function from R to R. Which of these are
one-to-one, and which are onto? Prove that your answer is correct.
(a) a(x) = 1
(b) b(x) = x
(c) c(x) = x2
(d) e(x) = 1/
homework 3 (Due in class, on February 4th)
(1) Let U = Z. Let A = cfw_1, 2, 3, 4, 5 and B = cfw_1, 3, 5, 7, 9. Specify each set
by listing its elements.
(a) cfw_a 2 A : a is even
(b) cfw_x 2 B : x is even
(c) cfw_a 2 A : a is odd
(d) cfw_b 2 B : b is odd
homework 2 (Due in class, on January 28th)
(1) (4.5) Write a two-column proof of each of the following deductions.
(a) P _ Q, Q _ R, Q, ) P &R.
(b) (E _ G) _ F, G&F, ) E.
(2) (4.9) Provide a justication for each of these proofs.
1
U =) V
2 U =) V
(a)
3
U
(1)
(2)
(3)
(4)
(5)
homework 5 (Due in class, on March 4th (Tuesday)
Suppose A, B, and C are sets (and U is the universal set as usual.) Prove
(a) Show A \ B = A \ (A \ B)
(b) Let X = A \ B and show A [ B = (A \ X) [ (B \ X) [ X.
(c) A \ B = A \ B
(d) A =
MATH 2000 - Midterm
February 11, 2014
Last Name:
First Name:
Student ID:
Tutorial Time:
No external aid is allowed. Cell phones should be turned o before the start of this exam.
This exam is 60 minutes long.
There are 10 pages and 6 questions in this exam