4+x=5
x + 1 = 5 if x = 1
p=
q=
p
q
~p
~q
~ (p
q) q
p ~q
p=
q=
r=
( p ~ q)
~ (p
p
q) =~ q
p
(p
~ p (q
q) (~ p
q)
p) =~ p (~ q ~ p)
x:x
x:x
x:x
F
F
x =2
x < 100
2
(S
G
(S
G
G)
F
G
G)
(S
S
F)
G
(use a general Venn Diagram for three sets A,B, and C
where all
REVIEW EXAM III
1.
Find the dual of (a + b) + a b
2.
Use De Morgans Law to find ( x( yz + y z )
3.
Obtain the complete sum of products for for:
a. y ( x + z )
b. x + y ( z + x y )
4.
Construct Karnaugh maps for the following expressions and find the minim
MAD1100: Discrete Math Spring 2016
Review
1) A bag contains 10 yellow, 5 blue, and 2 green balls. If three balls are selected
at random without replacement, what are the probabilities of the following
events?
a. A=Two green and one yellow ball ar
MAD1100
E.Philias
Section 5.1 Relations
1) Let A =cfw_ 1,2,3,4. Consider the following relation R on A:
R=cfw_ ( 1,1), (2,2), (2,3), (3,2), (4,2), (4,4)
Determine if the relation is a) reflexive b) symmetric c) transitive. If not, show why.
2) Consider
MAD1100
E.Philias
Chapter 5: Relations and Functions
Section 5.1 Relations
An ordered pair consists of two elements, one of which is designated the first element and the
other the second element. We write such a pair as (a, b).
Equality: Two ordered p
MAD1100
E.Philias
Section 4.1 Finite Probability
1) An experiment results in one of the following points: , , , , or .
a) Find P( ) if P( ) = .1 , P(
= .2 , P( ) = .1, and P( ) = .1.
b) Find P( ) if P( ) = P( ), P(
= .1, P( ) = .2, and P( ) = .1.
c) Find
MAD1100
E.Philias
SETS AND LOGIC
SECTION 2.1 : SETS AND
ELEMENTS
A set is a well-defined collection of objects. The objects in a set are called the elements or
members of the set.
Note: By well-defined we mean there is a rule that enables us to deter
E.Philias
MAD1100
Counting Methods
Section 3.2 Permutations
The Fundamental Counting
Principle
The Fundamental Counting Principle
Suppose a procedure consists of n stages. At the first stage there are there are !" choices, at
the second stage ther
MAD1100
E.Philias
Predicates
Consider the following statements: x > 3, x = y + 3, x + y = z
The truth value of these statements has no meaning without specifying the values of x, y, z.
However, we can make propositions out of such statements.
A predicat
E.PHILIAS
MAD1100
Section 3.2 & 3.3
Permutations & Combinations
1) A restaurant offers nine different desserts, which it serves with coffee, decaffeinated coffee, tea, milk, or
hot chocolate. In how many different ways can one order a dessert and a d
MAD1100
EXAM #1 Review
1) List the elements of a given set.
A=cfw_ x: x is a perfect square, -16 # 16
E=cfw_ x: x is an integer such that # $ = 3.1
2) Translate symbolic compound statement into words.
Let P represent the statement It is Tuesday and le
MAD 1100U02 (16312)
MATH IT (GL 132)
Spring 2015
Instructor: Abdel-Rida Saleh
Telephone: (305) 237-0332
Email: [email protected]
Office hours: T 4:00 pm 4:55 pm.
R 3:30pm 4:55 pm.
Office: DM 415B
Text: A First Course in Discrete Mathematics by John C. Mollu
REVIEWEXAMII
1.Determinewhetherthefollowingrelationson A = cfw_1, 2, 3, 4 arereflexive,
symmetricand/ortransitive?
a. cfw_(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)
b. cfw_(1,1),(2,2),(3,3)
c. cfw_(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)
2.Whichofthefollowingareequiv
Review Exam I - Solutions
1.
a.YES
2.
a. If I bought a lottery ticket this week, then I won the million-dollar jackpot.
b.NO
c.YES
b. If I didn't but a lottery ticket this week, then I didn't win the million-dollar
jackpot.
c. Either I buy a lottery ticke
4+x=5
x + 1 = 5 if x = 1
p=
q=
p
q
~p
~q
~ (p
q) q
p ~q
p=
q=
r=
( p ~ q)
~ (p
p
q) = ~ q
p
(p
~ p (q
q ) (~ p
q)
p ) = ~ p (~ q ~ p )
x:x
x:x
x:x
F
F
x =2
x < 100
2
(S
G
(S
G
G)
F
G
G)
(S
S
F)
G
(use a general Venn Diagram for three sets A,B, and C
where
REVIEW EXAM III
1.
Find the dual of (a + b) + a b
2.
Use De Morgans Law to find ( x( yz + y z )
3.
Obtain the complete sum of products for for:
a. y ( x + z )
b. x + y ( z + x y )
4.
Construct Karnaugh maps for the following expressions and find the minim
Solutions Supplementary Problems Predicates
1. a) There is a student who spends more than five hours every weekday in class.
b) All students spend more than five hours every weekday in class.
c) There is a student who does not spend more than five hours e
Review Exam I - Solutions
1.
a.YES
2.
a. If I bought a lottery ticket this week, then I won the million-dollar jackpot.
b.NO
c.YES
b. If I didn't but a lottery ticket this week, then I didn't win the million-dollar
jackpot.
c. Either I buy a lottery ticke
H D 1/00
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gamma [9A Mb
905- I.
. THvste, ML 41 Venn 13- 4ng #
(Afarseclian oqc all gas is n"w1:j)m8rm ( me a
4) (Maw, C = id
('13'497)? "(1 cfw_55/183
cfw_41:5 A cfw_max: cfw_53 ,.
b)(408)U(6/1cyr (and)
$1151! (
MAD1100
E.Philias
Section 2.1 Sets and Elements
1) List the elements from each set
a. A=cfw_ x: x is an odd integer, 5 " 93
b. B=cfw_ x: x is a perfect square, x 225
c. C= cfw_ y: y is a positive integer, y is a multiple of 3
d. D=cfw_ x: x is positive