Download Homework 1 Solution - Han

MATH 2001 Homework
Han-Bom Moon
Homework 1 Solution
Section 1.1 and 1.2.
• Do not abbreviate your answer. Write everything in full sentences.
• Write your answer neatly. If I couldn’t understand it, you’ll get 0 point.
• You may discuss with your classmates. But do not copy directly.
1. Let the following statements be given.
p =
“You are in Seoul.”
q =
“You can taste authentic Kimchi.”
r =
“You are in South Korea.”
(a) Translate the following statement into symbols of formal logic.
If you are not in South Korea, then you are not in Seoul or you cannot taste
authentic Kimchi.
¬r → ¬p ∨ ¬q
(b) Translate the following formal statement into English.
q → (r ∧ ¬p)
If you can taste authentic Kimchi, then you are in South Korea and you are
not in Seoul.
(c) Give the contrapositive of (b) in the symbols of formal logic and translate it
into everyday English.
¬(r ∧ ¬p) → ¬q (or (¬r ∨ p) → ¬q)
If you are not in South Korea or you are in Seoul, then you cannot taste
authentic Kimchi.
2. Consider the following statement p.
If 2 is a factor of x, then 6 is a factor of x.
(a) Give the converse of p.
If 6 is a factor of x, then 2 is a factor of x.
(b) Give the contrapositive of p.
If 6 is not a factor of x, then 2 is not a factor of x.
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MATH 2001 Homework
3.
Han-Bom Moon
(a) Use truth tables to show that p → q is not logically equivalent to its converse.
p
T
T
F
F
q
T
F
T
F
p→q
T
F
T
T
q→p
T
T
F
T
There are some cases that two statements have different truth value. Therefore they are not logically equivalent.
(b) Use truth tables to show that ¬(p ∨ q) is logically equivalent to ¬p ∧ ¬q.
p
T
T
F
F
q
T
F
T
F
p∨q
T
T
T
F
¬(p ∨ q)
F
F
F
T
¬p
F
F
T
T
¬q
F
T
F
T
¬p ∧ ¬q
F
F
F
T
The fourth column and the last column have the same truth value. So they
are logically equivalent.
(c) Use truth tables to show that ¬(p ∧ q) is logically equivalent to ¬p ∨ ¬q. ((b)
and (c) are called De Morgan’s law.)
p
T
T
F
F
q
T
F
T
F
p∧q
T
F
F
F
¬(p ∧ q)
F
T
T
T
¬p
F
F
T
T
¬q
F
T
F
T
¬p ∨ ¬q
F
T
T
T
The fourth column and the last column have the same truth value. So they
are logically equivalent.
4. If P → Q is true, then mathematicians say that “Statement P is a sufficient condition for statement Q”, and “Statement Q is a necessary condition for statement P ”.
In other words, in order to know that Q is true, it is sufficient to know that P is
true. And in order for P to be true, it is necessary that Q must be true.
(a) Consider the following two statements:
p =
x is a positive integer.
q =
x is a rational number.
Which one is a sufficient condition of the other?
If p is true, then q is true. Thus p → q is true. Therefore p is a sufficient
condition for q.
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MATH 2001 Homework
Han-Bom Moon
(b) Give a necessary condition on a quadrilateral A for A to be a rectangle.
A is a parallelogram. (Of course, there may be different answers.)
(c) Let n be an integer. Give a sufficient condition on n for n/2 be an even
integer.
n is a multiple of 8. (Another example is simply “n = 400 ).
5. Use truth tables to establish the following statement tautology.
(a) modus tollens.
p
T
T
F
F
q
T
F
T
F
¬p
F
F
T
T
¬q
F
T
F
T
p→q
T
F
T
T
(¬q) ∧ (p → q)
F
F
F
T
((¬q) ∧ (p → q)) → ¬p
T
T
T
T
The last column is always true. Therefore it is a tautology.
(b) Proof by division into cases:


p∨q 
⇒r
p→r

q→r 
p
q
r
p∨q
p→r
q→r
(p ∨ q) ∧ (p → r)
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
T
T
T
T
T
F
F
T
F
T
F
T
T
T
T
T
F
T
T
T
F
T
T
T
F
T
F
T
T
F
F
(p ∨ q) ∧ (p → r)
∧(q → r)
T
F
T
F
T
F
F
F
6. Write a proof sequence for the following assertion. Justify each step.


p

⇒ ¬q
p→r


q → ¬r
3
((p ∨ q) ∧ (p → r)
∧(q → r)) → r
T
T
T
T
T
T
T
T
MATH 2001 Homework
Han-Bom Moon
1.
2.
3.
4.
5.
6.
statement
p
p→r
q → ¬r
r
¬¬r
¬q
reason
given
given
given
modus ponens, 1 and 2
double negation, 4
modus tollens, 3 and 5
7. Write a proof sequence for the following assertion. Justify each step.
(p ∨ q) ∨ (p ∨ r) ⇒ ¬r → (p ∨ q)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
8.
statement
(p ∨ q) ∨ (p ∨ r)
(p ∨ q) → (p ∨ q) ∨ r
(p ∨ q) → r ∨ (p ∨ q)
(p ∨ r) → q ∨ (p ∨ r)
(p ∨ r) → (q ∨ p) ∨ r
(p ∨ r) → (p ∨ q) ∨ r
(p ∨ r) → r ∨ (p ∨ q)
r ∨ (p ∨ q)
¬¬r ∨ (p ∨ q)
¬r → (p ∨ q)
reason
given
addition
commutativity and 2
addition
associativity and 4
commutativity and 5
commutativity and 6
proof by division of cases, 1, 3, and 7
double negation and 8
implication and 9
(a) Write a simple paragraph which is using modus ponens in everyday English.
If Dr. Moon’s class is fun, then many students will retake his class. Dr.
Moon’s class is fun. Therefore Many students will retake his class.
(b) Write a simple paragraph which is using modus tollens in everyday English.
If Han-Bom is hungry, then he will eat all of sandwiches in his refrigerator. He doesn’t eat all of sandwiches in his refrigerator. Therefore he is not
hungry.
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