Laws of Logic in Discrete Structures and their applications

Applying Laws of Logic
 Using laws of logic simplify the statement form.
p  [~(~p  q)]
 Solution:
 p  [~(~p)  (~q)] DeMorgan’s Law
 p  [p(~q)] Double Negative Law
 [p  p](~q) Associative Law for 
 p  (~q) Indempotent Law
This is the simplified statement form.

EXAMPLE
 Using Laws of Logic, verify the logical equivalence.
~ (~ p  q)  (p  q)  p
Solution:
 (~(~p)  ~q) (p  q) DeMorgan’s Law
 (p  ~q)  (p  q) Double Negative Law
 p  (~q  q) Distributive Law in
reverse
 p  c Negation Law
 p Identity Law

SIMPLIFYING A STATEMENT:
 “You will get an A if you are hardworking and the sun
shines, or you are hardworking and it rains.”
 Solution:
Let
p = “You are hardworking’
q = “The sun shines”
r = “It rains”
 The condition is then (p  q)  (p  r)

(p  q)  (p  r)
  p  (q  r) Distributive law in reverse
 Putting p  (q  r) back into English, we can
rephrase the given sentence as
 “You will get an A if you are hardworking and the
sun shines or it rains.”

EXERCISE:
 Use Logical Equivalence to rewrite each of the following
sentences more simply.
 1. It is not true that I am tired and you are smart.
{I am not tired or you are not smart.}
 2. It is not true that I am tired or you are smart.
{I am not tired and you are not smart.}
 3. I forgot my pen or my bag and I forgot my pen or
my glasses.
4. It is raining and I have forgotten my umbrella, or it
is raining and I have forgotten my hat.

EXERCISE:
 Use Logical Equivalence to rewrite each of the following
sentences more simply.
 1. It is not true that I am tired and you are smart.
{I am not tired or you are not smart.}
 2. It is not true that I am tired or you are smart.
{I am not tired and you are not smart.}
 3. I forgot my pen or my bag and I forgot my pen or
my glasses.
{I forgot my pen or I forgot my bag and glasses.
 4. It is raining and I have forgotten my umbrella, or it
is raining and I have forgotten my hat.
{It is raining and I have forgotten my umbrella or my
hat.}

CONDITIONAL STATEMENT or
IMPLICATION
 Introduction
Consider the statement:
"If you earn an A in Math, then I'll buy you a
computer."
 This statement is made up of two simpler statements:
p: "You earn an A in Math," and
q: "I will buy you a computer."
if p is true, then q is true, or, more simply, if p, then q.
We can also phrase this as p implies q, and we write p
 q.

The original statement is then saying:
if p is true, then q is true
Or
more simply, if p, then q.
We can also phrase this as p implies q, and we write p
 q.

TRUTH TABLE for p  q
p q p  q
T T T
T F F
F T T
F F T

CONDITIONAL STATEMENTS OR
IMPLICATIONS:
 Definition:
 If p and q are statement variables, the conditional of
q by p is “If p then q” or “p implies q” and is denoted
p  q.
 It is false when p is true and q is false; otherwise it is
true.
 The arrow " " is the conditional operator
 and in p  q the statement p is called the
hypothesis
 (or antecedent)
 q is called the conclusion (or consequent).

PRACTICE WITH CONDITIONAL
STATEMENTS:
 Determine the truth value of each of the following
conditional statements:
“If 1 = 1, then 3 = 3.”
“If 1 = 1, then 2 = 3.”
“If 1 = 0, then 3 = 3.”
“If 1 = 2, then 2 = 3.”
“If 1 = 1, then 1 = 2 and 2 = 3.”
“If 1 = 3 or 1 = 2 then 3 = 3.”

PRACTICE WITH CONDITIONAL
STATEMENTS:
 Determine the truth value of each of the following
conditional statements:
“If 1 = 1, then 3 = 3.” TRUE
“If 1 = 1, then 2 = 3.” FALSE
“If 1 = 0, then 3 = 3.” TRUE
“If 1 = 2, then 2 = 3.” TRUE
“If 1 = 1, then 1 = 2 and 2 = 3.” FALSE
“If 1 = 3 or 1 = 2 then 3 = 3.” TRUE

ALTERNATIVE WAYS OF EXPRESSING
IMPLICATIONS
 The implication p  q could be expressed in many
alternative ways as:
“if p then q” “not p unless q”
“p implies q” “q follows from p”
“if p, q” “q if p”
“p only if q” “q whenever p”
“p is sufficient for q” “q is necessary for p”

EXERCISE:
 Write the following statements in the form “if p, then
q” in English.
a) Your guarantee is good only if you bought
your CD less than 90 days ago.
If your guarantee is good, then you must have
bought your CD less than 90 days ago.
b) To get tenure as a professor, it is sufficient to
be world-famous.
If you are world-famous, then you will get tenure
as a professor.

c) That you get the job implies that you have the
best credentials.
If you get the job, then you have the best
credentials.
d) It is necessary to walk 8 miles to get to the top
of the Peak.
If you get to the top of the peak, then you must
have walked 8 miles.

TRANSLATING ENGLISH SENTENCES TO
SYMBOLS:
 Let p and q be propositions:
p = “you get an A on the final exam”
q = “you do every exercise in this book”
r = “you get an A in this class”
 Write the following propositions using p, q, and r and
logical connectives.

 To get an A in this class it is necessary for you to get
an A on the final.
SOLUTION r  p
 You do every exercise in this book; You get an A on
the final, implies, you get an A in the class.
SOLUTION ?
 Getting an A on the final and doing every exercise in
this book is sufficient for getting an A in this class.
SOLUTION ?

 To get an A in this class it is necessary for you to get
an A on the final.
SOLUTION r  p
 You do every exercise in this book; You get an A on
the final, implies, you get an A in the class.
SOLUTION q  p  r
 Getting an A on the final and doing every exercise in
this book is sufficient for getting an A in this class.
SOLUTION p  q  r

TRANSLATING SYMBOLIC
PROPOSITIONS TO ENGLISH
 Let p, q, and r be the propositions:
p = “you have the flu”
q = “you miss the final exam”
r = “you pass the course”
 Express the following propositions as an English
sentence.

 p  q
 ~q  r
 ~p  ~q r

 p  q
If you have flu, then you will miss the final exam.
 ~q  r
If you don’t miss the final exam, you will pass the
course.
 ~p  ~q r
If you neither have flu nor miss the final exam, then
you will pass the course.

HIERARCHY OF OPERATIONS
FOR LOGICAL CONNECTIVES
~ (negation)
 (conjunction)
 (disjunction)
 (conditional)

Construct a truth table for the statement
form (p  ~ q)  ~ p

form (p  ~ q)  ~ p
p q ~q ~p p  ~q (p  ~ q)  ~ p
T T F F T F
T F T F T F
F T F T F T
F F T T T T

form (p q)(~ p  r)

form (p q)(~ p  r)
p q r pq ~p ~pr (p  q)  (~ p r)
T T T T F T T
T T F T F T T
T F T F F T F
T F F F F T F
F T T T T T T
F T F T T F F
F F T T T T T
F F F T T F F

LOGICAL EQUIVALENCE INVOLVING
IMPLICATION
 Use truth table to show p  q  ~q  ~p

LOGICAL EQUIVALENCE INVOLVING
IMPLICATION
 Use truth table to show p  q  ~q  ~p
p q ~q ~p p  q ~q  ~p
T T F F T T
T F T F F F
F T F T T T
F F T T T T
same truth values

IMPLICATION LAW
 p  q  ~p  q

IMPLICATION LAW
 p  q  ~p  q
p q pq ~p ~p  q
T T T F T
T F F F F
F T T T T
F F T T T

NEGATION OF A CONDITIONAL
STATEMENT
 Since p  q  ~p  q therefore
 ~ (p  q)  ~ (~ p  q)
 ~ (~ p)  (~ q) by De Morgan’s law
 p  ~ q by the Double Negative law
 Thus the negation of “if p then q” is logically
equivalent to “p and not q”.
 Note:
Accordingly, the negation of an if-then statement
does not start with the word if.

EXAMPLES
 Write negations of each of the following
statements:
 If Ali lives in Pakistan then he lives in Lahore.
Ali lives in Pakistan and he does not live in Lahore.
 If my car is in the repair shop, then I cannot get to
class.

EXAMPLES
 If Ali lives in Pakistan then he lives in Lahore.
Ali lives in Pakistan and he does not live in Lahore.
 If my car is in the repair shop, then I cannot get to
class.
My car is in the repair shop and I can get to class.

Write negations of each of the following
statements:
 If x is prime then x is odd or x is 2.
 If n is divisible by 6, then n is divisible by 2 and n is
divisible by 3.

 If x is prime then x is odd or x is 2.
x is prime but x is not odd and x is not 2.
 If n is divisible by 6, then n is divisible by 2 and n is
divisible by 3.
n is divisible by 6 but n is not divisible by 2 or by 3.

INVERSE OF A CONDITIONAL
STATEMENT
 The inverse of the conditional statement p  q is
~p  ~q
 A conditional and its inverse are not equivalent as
could be seen from the truth table.

INVERSE OF A CONDITIONAL
STATEMENT
 The inverse of the conditional statement p  q is
~p  ~q
 A conditional and its inverse are not equivalent as
could be seen from the truth table.
p q p  q ~p ~q ~p ~q
T T T F F T
T F F F T T
F T T T F F
F F T T T T
different truth values in rows 2 and 3

WRITING INVERSE
 If today is Friday, then 2 + 3 = 5.
If today is not Friday, then 2 + 3  5.
 If it snows today, I will ski tomorrow.
 If P is a square, then P is a rectangle.
 If my car is in the repair shop, then I cannot get
to class.

WRITING INVERSE
If today is not Friday, then 2 + 3  5.
If it does not snow today I will not ski tomorrow.
 If P is a square, then P is a rectangle.
If P is not a square then P is not a rectangle.
 If my car is in the repair shop, then I cannot get
to class.
If my car is not in the repair shop, then I shall
get to the class.

CONVERSE OF A CONDITIONAL
STATEMENT
 The converse of the conditional statement p  q is
q p
 A conditional and its converse are not equivalent.
 i.e.,  is not a commutative operator.
p q p  q q  p
T T T T
T F F T
F T T F
F F T T
not the same

WRITING CONVERSE
If 2 + 3 = 5, then today is Friday.
to class.

WRITING CONVERSE
If 2 + 3 = 5, then today is Friday.
I will ski tomorrow only if it snows today.
If P is a rectangle then P is a square.
to class.
If I cannot get to the class, then my car is in the
repair shop.

CONTRAPOSITIVE OF A CONDITIONAL
STATEMENT
 The contrapositive of the conditional statement p 
q is ~ q  ~ p
 A conditional and its contrapositive are equivalent.
Symbolically p  q  ~q  ~p

If 2 + 3  5, then today is not Friday.
to class.

WRITING CONVERSE
If 2 + 3  5, then today is not Friday.
I will not ski tomorrow only if it not snows today.
If P is not a rectangle then P is not a square.
to class.
If I can get to the class, then my car is not in the
repair shop.

Laws of Logic in Discrete Structures and their applications

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