1) [10 points] Let:
R(x, y ): x is related to y
S (x): x is single
H (x): x is happy
Using the statements above, analyze the statement [which does not need to be true!]: Every
happy person is related to a person that is single.
x, H (x) (y st R(
1., 2., 3. Do page 186/187 #4, 5, 16
4. Let A and B be sets and let R and S be relations from A to B .
Prove: If S R, then S 1 R1 .
5. Suppose R is a relation on the set A. In each case prove the
statement or give a counterexa
page 201 # 20 and
Let R be a partial order relation on the set A, and let B A. Recall
that R1 also is a partial ordering on A. For example, if R is , then
R1 is .
1. Suppose b B . Show that B is the R-smallest element of B , i
1. Let R and S be equivalence relations on the set A. (a) Show that
R S ( x A [x]R [x]S ).
(b) Show that
R S (F A/R G A/S (F G).
2. Let R be a partial ordering on the set
A and let B A. Show that if b B is
a lower bound for B ,
Math 307, Fall 2013
Introduction to Abstract Mathematics
Instructor: Dr. Stefan Richter, 322 Ayres Hall, Tel.: 974-4286
e-mail: Richter at math dot utk dot edu
Office hours: MWF 1:15-2:00 & by appointment
Section, Time & Place: 43593, MWF 12:20-1:10, Ayre
1) [8 points] Rewrite the statement [about real numbers]:
[x R, y N st [(x y ) (x + y > 0) (x = y + 2)]
as a positive statement [without the symbol].
x R st y N, [(x y ) (x + y 0) (x = y + 2)]
2) [8 points] Fill the truth table below.
1) [15 points] Suppose that A, B and C are sets such that A B \ C . Show that A C = .
Proof. Suppose that x A C . [We need to derive a contradiction!] Then x A and x C .
But, since x A and A B \ C , we have that x B \ C , i.e., x B and x C . But we