MAT 243 WRITTEN HOMEWORK 1
NAME: Solutions
(1) Fill in the blank in the statements below:
(a) Two propositions are logically equivalent if and only if they have the same truth values.
(b) A tautology is a a proposition that is always true.
(c) The negatio
MAT 243 HW 3 SOLUTIONS
(1) (4 pts) Fill in the blank in the statements below:
(a) P (S) is the set containing all subsets of S i.e., P (S) = cfw_X | X S
(b) A B = cfw_x | x A x B
/
(c) A function f : A B is one-to-one if and only if a A b A (f (a) = f (b)
Mat 243 Ionascu Test II Review
SHORT STORY of the TOPICS to be covered on Test II
Section 2.3 Functions: understand the definition of a function, domain, codomain, range; what it mean for
a function to be injective (or one-to-one), surjective (or onto) an
Assignment Unit1 Propositional Logic
1. (1 pt)
Enter T for each true proposition, F for each false proposition
and N for each statement which is not a proposition.
1. What time is it?
2. x+y=y+x for every pair of real numbers x and y.
3. All insects are a
MAT 243 ONLINE WRITTEN HW 6 SOLUTIONS
(1) (4 pts) Fill in the blank in the statements below:
(a) When making a sequence of choices in counting we are using the Product Rule.
(b) We are using the Sum Rule when choosing among mutually exclusive alternatives
Solutions to exam 1
1. Construct the truth table for the following proposition:
(p q ) (p r)
Solution:
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
p q p r p r (p q ) (p r)
T
FF
T
T
T
FT
F
F
F
FF
T
T
F
FT
F
T
T
TF
F
F
T
TT
T
T
T
TF
F
F
T
TT
T
T
2
Assignment Unit2 Propositional Equivalence
1. (1 pt) Complete the following truth table by filling in the
blanks with T or F as appropriate.
p q p!q :p :q :q!:p
TT
TF
FT
FF
p!q and :q!:p are
_ A. not logically comparable
_ B. not logically equivalent
_ C.
MAT 243 ONLINE WRITTEN HW 7 SOLUTIONS
(1) (4 pts) Fill in the blank in the statements below:
(a) The general form of the Inclusion-exclusion formula says that |A1 A2 . An | =
n
|Ai |
=
i=1
|Ai Aj Ak | . + (1)n+1 |A1 A2 . An |
|Ai Aj | +
1i<jn
1i<j<kn
(b)
KUNAL LANJEWAR
Base Case:
Every member of the initial population cfw_0 is even.
Inductive Step:
We want to show that both 3 n+2 and n2 are even. Now suppose that
n is even. Then, 3 + 2 and n2 are also even, by definition of even
numbers. Therefore, by the
KUNAL LANJEWAR
1. There are 25 prime numbers in the [1,100] interval.
Let total number of prime number = n.
Therefore n = 25.
Probability of n is not prime =
10025 75 3
=
= =0.75
100
100 4
2. Let,
T = Total number of people = 100
C = Cat owners = 52
D = D
KUNAL LANJEWAR
Base Case:
Every member of the initial population cfw_0 is even.
Inductive Step:
We want to show that both 3 + 2 and ! are even. Now suppose that
is even. Then, 3 + 2 and
Nicole Johnson
1.
1) Counting Principal
2) mutually exclusive
3) ordered list
4) if there are k boxes and you place n objects into them then at least one box will
2.
contain at least
) (10,6) = 10 9 8 7 6 5
b) 59 6! =
) 85 6! =
) 84 6! =
845! 2! =
Edward Beickman
Beickman BC Calculus 2012-2013
WeBWorK assignment number Set Theory Problems is due : 07/08/2013 at 02:43pm MST.
Go to Mr. Beickmans Home Page ( http:/ww2.chandler.k12.az.us/Domain/4854) for the course syllabus, grading policy and
other in
KUNAL LANJEWAR
1.
Let,
P be the set of all people
L(x) be the predicate "X likes ice cream"
K(x) be the predicate "X is a kid"
i.
Every kid loves ice cream Premise
x D, K (x) L (x)
ii.
Joey
KUNAL LANJEWAR
1.
Let,
P be the set of all people
L(x) be the predicate "X likes ice cream"
K(x) be the predicate "X is a kid"
i.
Every kid loves ice cream Premise
x D, K (x) L (x)
ii.
Joey doesnt love ice cream Premise
L (Joey)
iii.
Therefore, Joey is
KUNAL LANJEWAR
1.
S=cfw_1
cfw_, cfw_ 1
2.
3.
Suppose, for any S [1,2] we have b1[0,1] and
f ( S1 )=( b1 )+ 1=S . Therefore is Onto or bijective function.
4.
Here domain lies between (1, ) and codomain lies between (1, )
and range is between (2, ) . Sin
KUNAL LANJEWAR
1. D
2. False It is not always true that summation of two functions of same order to have same
order. If,
f 1 ( x )=Og ( x ) , f 2 ( x )=Og ( x ) thenf 1 ( x )+ f 2 ( x ) maymay not be same as Og ( x ) .
Eg.
f 1 ( x )=xf 2 ( x )=x then both
MAT 243 ONLINE WRITTEN HW 6 SOLUTIONS
(1) (4 pts) Fill in the blank in the statements below:
(a) When making a sequence of choices in counting we are using the Product Rule.
(b) We are using the Sum Rule when choosing among mutually exclusive alternatives
MAT 243 ONLINE WRITTEN HW 5 SOLUTIONS
NAME:
(1) (4 pts) Fill in the blank in the statements below:
(a) In an inductive proof verifying the condition for n = 1 (or the lowest possible value)
is called the base step or base case.
(b)-(c) In the induction st
MAT 243 ONLINE WRITTEN HW 4
NAME:
(1) (4 pts) Fill in the blank in the statements below:
(a) A function f (x) is big-O of g(x) if and only if there exists constant C and k such that
|f (x)| C|g(x)| for all x > k.
(b) An integer a divides integer b if and
MAT 243 EXAM 1 PROOFS SOLUTION
On this assignment, you are expected to write two clear proofs, i.e. an argument with complete, grammatically correct English sentences, not a formal argument. Correct logical
structure is essential for a full score. You do
Nicole Johnson
1.
1) If a conclusion follows logically from the premises
2) Modus Ponens
3) The Fallacy of affirming the conclusion
4) A theorem that can be established directly from s theorem that has already been
proved.
2.
a.
1)
2) s premise
3) q r pr
MAT 243 HW 2 SOL
Exercise #1.
(1) the conclusion follows logically from the premises, i.e. if the conclusion must be true
given that the premises are true.
(2) Modus ponens
(3) Fallacy of assuming the conclusion.
(4) theorem that is an immediate consequen
Nicole Johnson
1.
1) IFF there are constants C and n0 , such that |()| |()| whenever x > n0
2) If there is an integer C such that b=ac, or if is an integer
3) IFF there is an integer k such that a = b + km
4) relatively prime
2.
?
3. fast modular exponent
MAT 243 ONLINE WRITTEN HW 6 SOLUTIONS
(1) (4 pts) Fill in the blank in the statements below:
(a) When making a sequence of choices in counting we are using the Product Rule.
(b) We are using the Sum Rule when choosing among mutually exclusive alternatives
MAT 243
Spring 2015
Homework 1
Due 01/27/2015
Name :
1. Let p and q be the propositions
p : It is below freezing.
q : It is snowing.
Write these propositions using p, q, and logical connectives (including negations).
It is below freezing and snowing.
It i
MAT 243 Spring 2015
Review for Test 1
1. Which of the following sets are equal to the set of all integers that are even. There
may be more than one or none.
(a) cfw_2n|n R
(b) cfw_2n|n Z
(c) cfw_n Z|n = 2k and k Z
(d) cfw_2n
(e) cfw_0, 2, 4, 6, . . .
2.
MAT 243 HW 3 SOLUTIONS
(1) (4 pts) Fill in the blank in the statements below:
(a) P (S) is the set containing all subsets of S i.e., P (S) = cfw_X | X S
(b) A B = cfw_x | x A x
/ B
(c) A function f : A B is one-to-one if and only if a A b A (f (a) = f (b
Nicole Johnson
1.
1) IFF there are constants C and n0 , such that |()| |()| whenever x > n0
2) If there is an integer C such that b=ac, or if is an integer
3) IFF there is an integer k such that a = b + km
4) relatively prime
2.
?
3. fast modular exponent
MAT 243 WRITTEN HOMEWORK 1
(1) Fill in the blank in the statements below:
(a) Two propositions are logically equivalent if and only if they have the same truth values.
(b) A tautology is a proposition that is always true
(c) The negation of if p then q is
Nicole Johnson
1.
1) Basis Induction
2) Inductive hypothesis
3) Natural NUmbers
4) Equation which defines a sequence based on a rule which gives next term as a
function of the previous term.
2.
(): 1 + 5 + + (4 3) = 22
Base Case: n=1
(1): 1 = 2(1)2 1
1 =
MAT 243 HW 3 SOLUTIONS
(1) (4 pts) Fill in the blank in the statements below:
(a) P (S) is the set containing all subsets of S i.e., P (S) = cfw_X | X S
(b) A B = cfw_x | x A x
/ B
(c) A function f : A B is one-to-one if and only if a A b A (f (a) = f (b