Unit-1: Logic and Proof
Proposition: Any information or any declaration that is
either true or false but not both, known as propositional
statement or a simply we call proposition.
Examples of Proposition:
1. Delhi is the capital of India.
2. Delhi is not the capital of India.
3. 1-1=0.
Examples Continued…
4. 2=5
5. I am fine.
6. Today is the raining day.
7. Dr. Pankaj is a Mathematics teacher.
Examples of Not Propositions
1. y-1=0.
(It is True for y=1 and False for y=2)
2. x+y=z.
(It is True for x=1, y=1, z=2 and False for x=1,
y=1, z=5)
3. What is the answer? (It is a question, not the
Information)
Mixed Examples
1. Read the statement carefully. (It is an instruction, not
information, so not proposition)
2. Please tell me some information. (It is a request, not
information, so not proposition)
3. You should work hard to pass the exam. (It is an
advice, not information, so not proposition)
4. Mr. A and Mr. B are not friends. (proposition)
5. Today is Sunday. (proposition)
Time for Polls
Question:
Are you able to understand the Proposition?
Options:
A. Yes
B. No
Time for Puzzle
Question: Which of the following statements is a
proposition?
• The set A={(x,y): x^2+y^2=1, x,y real numbers}
is equation of circle.
• Always be careful
• x+y+1=0, x,y integers, is linear equation.
Important Points
Note: Propositions are denoted by small letters
as p, q, r, s, t…
Note: If any proposition gives true information
then we denote it by T and if give false
information then denoted by F.
Note: These both values {T, F} are called the
truth values of the propositions.
Compound Proposition
If two or more statements are combine and
make another new statement then new
statement is called compound statement.
Example: Let us consider two propositions
p: Today is Sunday
And
q: Today is the raining day.
Then we have following compound statements
Examples
1. Today is Sunday and today is the raining day.
2. Today is Sunday or today is the raining day.
3. Today is Sunday but Today is not the raining
day.
4. Today is Sunday and today is the raining day
but it is sun shine.
Negation of the statement
Let p be the proposition then negation of p is denoted by
–p and it is the opposite meaning of the proposition.
To represent the negation of a proposition we write: It is
not the case that (given statement p)
Example: Let the proposition is
p: Today is Friday.
Then negation is
-p: It is not the case that today is Friday
-p: Today is not Friday.
Example continued…
Example: Let the proposition is
p: I have at least two mobile phones.
Then negation is
-p: I have less than two mobile phones.
Example: Let the proposition is
q: I have two mobile phones.
Then negation is
-q: I have not two mobile phones.
Truth Table for Negation
p -p
T F
F T
Conjunction
Conjunction is used for the word and. It is
denoted by the symbol ^. For two propositions
p and q we write p^q.
Example: Let the two propositions are
p: Today is Friday.
q: It is raining today.
Then conjunction proposition p^q is
p^q: Today is Friday and It is raining today.
Truth Table for Conjunction p^q
p q p^q
T T T
T F F
F T F
F F F
Example
Example: Let the two propositions are
p: Delhi is the capital of India. (True)
q: 2+2=4 (True)
Then p^q is true statement because p and q
both statements are true.
Example Continued…
Example: Let the two propositions are
p: Delhi is the capital of India. (True)
q: 2=5 (False)
Then p^q is false statement because p and q
both statements are not true.
Example Continued…
Example: Let the two propositions are
p: Punjab is the capital of India. (False)
q: 2=5 (False)
Then p^q is false statement because p and q
both statements are not true.
Puzzle
Question 1: The proposition p^q is true or false:
p: 3 is a prime number.
q: 2 is an even number.
Question 2: The proposition p^q is true or false
if
p: The smallest prime number is 1.
q: |x| is always positive for any real value of x.
Disjunction
Disjunction is used for the word or and denoted by
the symbol v. For two propositions p and q we write
pvq and read it p or q.
Example: Let the two propositions are
p: Today is Friday.
q: It is raining today.
Then disjunction proposition is
pvq: Today is Friday or It is raining today.
Truth Table for Disjunction pvq
p q pvq
T T T
T F T
F T T
F F F
Example
Example: Let the two propositions are
p: Delhi is the capital of India. (True)
q: 2+2 ≠4 (False)
Then pvq is true statement because p and q
both statements are not false.
Example Continued…
Example: Let the two propositions are
p: Delhi is not the capital of India. (False)
q: 2=5 (False)
Then pvq is false statement because p and q
both statements are false.
Puzzle
Question 1: The proposition pvq is true or false:
p: 3 is a prime number.
q: 2 is an even number.
Question 2: The proposition pvq is true or false:
p: The smallest prime number is 1.
q: |x| is not positive for some real value of x.
Puzzle
Consider the three propositions:
p: n+1>1 for all integer n.
q: The closed interval [0, 1] is a finite set with two
elements {0, 1}.
r: The open interval (0, 1) is infinite set of finite
lenght.
Then check whether the following are true or
false:
(i) p^q (ii) p^r (iii) q^r
(iv) pvq (v) pvr (vi) qvr
Exclusive Disjunction p⨁q
It is particular case of the disjunction or and
called exclusive or. It is denoted by p⨁q for
two propositions p and q.
Exclusive or is true if exactly one proposition is
true otherwise falls.
Note: If all the propositions are true or all are
falls then p⨁q is false.
Truth Table for Exclusive Disjunction p⨁q
p q p⨁q
T T F
T F T
F T T
F F F
Examples
Example 1: Let the two propositions are
p: 2 is smallest prime. (True)
q: 2+2 ≠4 (False)
Then p⨁q is true statement because exactly
one proposition p true.
Example 2: Let the third proposition is
r: 2+2=4 (True)
Then p⨁q is false; p⨁q is false, statement
because more than one statement is true.
Example Continued…
Example: Let the two propositions are
p: 2 is smallest prime. (True)
q: 2+2=4 (True)
Then p⨁q is false statement because more than
one statement is true.
Puzzle
Question 1: Check whether the proposition
p⨁q is true or false if
p: 3 is a prime number.
q: 2 is an even number.
Question 2: Check whether the proposition
p⨁q is true or false if
p: The smallest prime number is 1.
q: |x| is not positive for some real value of x.
Conditional Statement
It is denoted by the symbol → . For two
propositions p and q we write p →q and read it p
implies q or if p then q. Conditional statement is
false if p is true and q is false.
Example: Let the two propositions are
p: Delhi is not the capital of India. (False)
q: 2=5 (False)
Then pvq is false statement because p and q both
statements are false.
Example
Example: Let the two propositions are
p: Delhi is the capital of India. (True)
q: 2=5 (False)
Then p → q is false statement because p is true
and q is false.
Example Continued…
Example: Let the two propositions are
p: 1 is the smallest prime. (False)
q: 2+2 =4 (True)
Then p → q is true statement because the
statement p is true and q is false not
satisfied.
Truth Table for Conditional Statement p → q
p q p→q
T T T
T F F
F T T
F F T
Puzzle
Question 1: Check whether the proposition
p → q is true or false if
p: 3 is a prime number.
q: 2 is an even number.
Question 2: Check whether the proposition
p → q is true or false if
p: The smallest prime number is 1.
q: |x| is not positive for some real value of x.
Puzzle
Consider the three propositions:
p: n+1>1 for all positive integer n.
q: The closed interval [0, 1] is a finite set.
r: The open interval (0, 1) is a set.
Then check whether the following are true or
false:
(i) p → q (ii) p → r (iii) q → r
(iv) q → p (v) r → p (vi) r → q
Unless
This is used for negative sense. For two
propositions p and q we write p unless q. it is
false if both statements p and q are false. It is
also the same as -q → p.
For constructing the truth table for p unless q
we use the truth table of -q → p.
Truth Table for unless (p unless q)
(-q → p)
p q -q p unless q
T T F T
T F T T
F T F T
F F T F
Example of Unless
Example: Let the two propositions are
p: I will go out side for playing.
q: it is raining.
Then p unless q is the proposition
p unless q: I will go out side for playing unless it is
raining.
Meaning: If it is not raining then I will go out side for
playing .
Important Points
Note: If it is not raining then I will go out side for
playing .
Note: I will not go out side for playing if it is
raining.
Note: Above both statements are different
mathematically.
First statement represents -q → p
whereas second statement represent q → -p.
Puzzle
Question 1: Check whether the proposition
p unless q is true or false if
p: 3 is a prime number. (True)
q: 1 is an even number. (False)
Question 2: Check whether the proposition
p unless q is true or false if
p: The smallest prime number is 1.
q: |x| is not positive for some real value of x.
Converse, Inverse and contrapositive
Statements
Let p and q be two propositions then conditional
statement is p → q and
its converse is denoted by q → p.
its inverse is denoted by -p → -q.
its contrapositive is denoted by -q → -p.
Truth table for Converse (q→ p)
p q q→ p
T T T
T F T
F T F
F F T
Truth table for Inverse (-p→-q)
p q -p -q -p→-q
T T F F T
T F F T T
F T T F F
F F T T T
Truth table for Contrapositive (-q→-p)
p q -q -p -q→-p
T T F F T
T F T F F
F T F T T
F F T T T
Example
Example: Let the two propositions are
p: 1 is the smallest prime. (False)
q: 2+2 =4 (True)
Then converse is false as q →p has the false truth value.
And the inverse is calculated as:
Since p is false which implies –p is true and similarly q is
true hence –q is false. Finally we get inverse -p →q is
false.
And the contrapositive is true.
Puzzle
Question 1: Check whether the proposition
contrapositive is true or false if
p: 3 is a prime number.
q: 1 is an even number.
Question 2: Check whether the proposition
inverse is true or false if
p: The smallest prime number is 1.
q: |x| is not positive for some real value of x.
