Math 11
Spring 2012
Name:
SID:
Quiz 2
Problem 2 on back
1. (5 points) Recall on Quiz 1 we dened the nite ordinals by
0=
1 = 0 cfw_0 = cfw_ = cfw_ = cfw_0
2 = 1 cfw_1 = cfw_ cfw_ = cfw_, cfw_ = cfw_0, 1
.
.
.
n + 1 = n cfw_n = cfw_0, 1, ., n
.
.
.
If P (X
Math 11
Spring 2012
Name:
SID:
Quiz 3
1. (5 points) Let X = cfw_a, b, c and Y = cfw_a, b. What is X Y ?
Proof.
X Y = cfw_(a, a), (a, b), (b, a), (b, b), (c, a), (c, b).
2. (5 points) Let A = cfw_1, 2, 3, 4 and R = cfw_(1, 1), (1, 2), (1, 3), (1, 4), (2, 3
Math 11
Spring 2012
Name:
SID:
Quiz 1
1. (6 points) Let X and Y be sets and |X |, |Y | denote the number of elements in X and
Y respectively (which we will assume are natural numbers). Suppose |X | = n, |Y | = m,
and |X Y | = k. What is
(a) |X Y |
(b) |X
~
Tuesday, January 17, 2017
8:02 PM
1.1 Introduction to Sets
Set: collection of things
Elements: things in the collection of a set
Expressed by: cfw_2,4,6,8
Infinite set: infinitely many elements; otherwise, it is called finite
set
Two sets are equa
Introduction
What is discrete mathematics?
In calculus we are concerned with continuity. No matter how close two
points on the real axis are it is always possible to insert a point between
them. Every segment can be divided indefinitely etc. For example,
Math 11
Spring 2012
Name:
SID:
Quiz 9
(Fantastic and Fun Problems for Your Entertainment on Back)
1. (5 points) A fair coin is tossed 5 times. Let A be the event that heads shows up
exactly once or tails shows up exactly once. What is P (A)? (For example,
Math 11
Spring 2012
Name:
SID:
Quiz 4
1. (4 points) Let C = cfw_1am, 2am., 11pm, 12pm bet the set of all hours on a clock and
C 2 be the relation dened by g xm h ym (where x, y cfw_a, p) i g = h. For
example 11 am 11 pm. Is this an equivalence relation?
Math 11
Spring 2012
Name:
SID:
Quiz 7
1. (5 points) A community consists of 10 mothers, each of whom has 3 children. If one
mother is to be selected as mother of the year and one child is to be selected as child
of the year in a given year, how many possi
Quiz 4 Solutions
Determine whether each of these arguments is valid. If an argument is correct, what rule
of inference is being used? If it is not valid, explain why.
Section 2 (11-noon):
1. If n is an even integer, then n2 + n +1 is odd. Suppose n2 + n +
Quiz 5 Solutions
Section 2 (11-noon):
1. Compute the number of reorderings of the word mathematics.
Solution. There are 11 letters in the word, but we do not distinguish between the
2 ms, the 2 as, and the 2 ts. Hence we have
11!
.
2! 2! 2!
2. Compute the
Quiz 2 Solutions
Section 2 (11-noon):
1. Let A = cfw_1, 2, 3, 4 and R = cfw_(1, 1), (2, 2), (3, 1), (3, 3), (4, 1), (4, 4) be a binary
relation on A.
(a) Draw the digraph of R.
1^
O
2
4
3
(b) Determine if R is reexive, symmetric, or transitive.
It is reex
Math 11
Spring 2012
Name:
SID:
Quiz 5
1. (5 points) Show that (p q ) and p q are logically equivalent.
Proof.
p q p q p q (p q ) p q
00 1 1
0
1
1
01 1 0
0
1
1
0
1
1
10 0 1
11 0 0
1
0
0
2. (5 points) Let P be the statement Andrew writes easier quizzes and
Math 11
Spring 2012
Name:
SID:
Quiz 8
Write answers at least in terms of factorials.
1. (5 points) How many rearrangements are there of the following words?
(a) assessment
(b) dfsaidafs
Solution.
answer is
(a) Here we have 10 letters with the 4 ss and 2 e