2. Proposition ( Statement)
A proposition is a declarative sentence that is either
true or false , but not both .
Example : 1. Delhi is capital of India.
Example : 2. 1+3 = 4
Example : 3. 3+3 = 5
Example : 4. Normal glass is unbreakable .
Proposition in example 1 and 2 are true.
Proposition in example 3 and 4 are false.
3. Examples that are not proposition .
1. Read this carefully
2. ๐ + ๐ = ๐
these are not declarative sentences .
Conventional letters ๐, ๐, ๐, ๐ are used for propositional
variables.
Truth value of a proposition is true and is denoted by T,
if it is true proposition.
Truth value of a proposition is false and is denoted by F,
if it is false proposition. .
4. Symbols and names used in
propositions
Symbol Name
~ Negation
ห Conjunction
ห Disjunction
โ Conditional
โ Biconditional
๊ Exclusive
5. Definition 1. Negation
Let ๐ be a proposition . The negation of ๐ is denoted
by ~๐
Example find the negation of the proposition โ today
is Mondayโ and express this in simple english.
Answer . โToday is not Mondayโ
or โIt is not Monday todayโ
Truth Table for the Negation of a Proposition
๐ ~๐
T F
F T
6. Definition 2. Conjunction
Let ๐ and ๐ be propositions . The conjunction of ๐
and ๐ denoted by ๐ห๐ , is the propostion โ๐ and ๐โ.
The conjunction ๐ห๐ is true when both ๐ and ๐ are
true and false otherwise.
Truth table for conjunction of two propositions
๐ ๐ ๐ห๐
T T T
T F F
F T F
F F F
7. Example 1. Find the conjunction of propositions ๐
and ๐ where ๐ is proposition โtoday is Mondayโ
and ๐ is proposition โ It is a raining day.โ
Solution: The conjunction of these propositions,
๐ห๐ is the proposition โToday is Monday and it is
raining todayโ this proposition is true on rainy
Monday and is false on any day that is not Monday
and on Monday when it is not rain.
8. Definition 3. Disjunction
Let ๐ and ๐ be propositions . The disjunction of ๐
and ๐ denoted by ๐ห ๐, is the propostion โ๐ or ๐โ.
The disjunction ๐ห ๐ is false when both ๐ and ๐ are
false and true otherwise.
Truth table for disjunction of two propositions
๐ ๐ ๐ห ๐
T T T
T F T
F T T
F F F
9. Definition 4. Exclusive
Let ๐ and ๐ be propositions . The exclusive or of ๐
and ๐ denoted ๐๐ฆ ๐๊๐ is the propostion that is true
when exactly one of ๐ and ๐ is true and is false
otherwise.
Truth table for the exclusive or of two propositions
๐ ๐ ๐๊๐
T T F
T F T
F T T
F F F
10. Definition 5. Conditional
Let ๐ and ๐ be propositions . The conditional statement
๐ โ ๐ is the proposition โ If ๐ then ๐" the conditional
statement ๐ โ ๐ is false when p is true and q is false , and
true otherwise .In conditional statement ๐ โ ๐, ๐ is called
the hypothesis and q is called the conclusion.
Truth table for the conditional statement ๐ โ ๐
๐ ๐ ๐ โ ๐
T T T
T F F
F T T
F F T
11. Example 2 .
Let ๐ be the statement โ Agrima learns discrete
mathematicsโ and ๐ the statement โAgrima will find a
good job.โ Express the statement ๐ โ ๐ as a statement in
English.
Solution : from definition of conditional statement
โ If Agrima learns discrete mathematics, then she will find
a good jobโ.
Or
โAgrima will find a good job when she learns discrete
mathematics.โ
12. Definition 6. Biconditional
Let ๐ and ๐ be propositions . The biconditional statement
๐ โ ๐ is the proposition โ ๐ if and only ๐" the
biconditional statement ๐ โ ๐ is true when p and q have
the same truth values, and false otherwise. Biconditional
statements are also called bi-implications .
Truth table for biconditional statement ๐ โ ๐
๐ ๐ ๐ โ ๐
T T T
T F F
F T F
F F T
13. Example 3. let p be the statement โyou can take the flightโ
and let q be the statement โ you buy a ticketโ.
Solution.
The ๐ โ ๐ statement is
You can take the flight if and only if you can buy a ticketโ .
Note the truth values of ๐ โ ๐ has the same truth values
as (๐ โ ๐) ห (๐ โ ๐)
Truth table of ๐ โ ๐ and (๐ โ ๐) ห (๐ โ ๐)
๐ ๐ ๐ โ ๐ ๐ โ ๐ ๐ โ ๐ (๐ โ ๐) ห (๐ โ ๐)
T T T T T T
T F F F T F
F T F T F F
F F T T T T
14. Definition 7. Converse
For conditional statement ๐ โ ๐
The proposition ๐ โ ๐ is converse of ๐ โ ๐ .
Truth table for converse of ๐ โ ๐
๐ ๐ ๐ โ ๐ ๐ โ ๐
T T T T
T F F T
F T T F
F F T T
15. Definition 8. Contrapositive
for conditional statement ๐ โ ๐
The proposition ~๐ โ ~๐ is contrapositive of ๐ โ
๐.
Note : same truth vales for the contrapositive of
๐ โ ๐
Truth table for Contrapositive of ๐ โ ๐
๐ ๐ ~๐ ~๐ ๐ โ ๐ ~๐ โ ~๐
T T F F T T
T F T F F F
F T F T T T
F F T T T T
16. Definition 9. Inverse
for conditional statement ๐ โ ๐ The proposition
~๐ โ ~๐ is inverse of ๐ โ ๐.
Truth table for Inverse of ๐ โ ๐
๐ ๐ ~๐ ~๐ ๐ โ ๐ ~๐ โ ~๐
T T F F T T
T F F T F T
F T T F T F
F F T T T T
17. Definition 10. Tautology
A compound proposition that is always true, no matter
what the truth values of the propositions that occur in it, is
called tautology.
Truth table of tautology
๐ ~๐ ๐ ห ~๐
T F T
F T T
18. Definition 10. Contradiction
A compound proposition that is always false, no
matter what the truth values of the propositions
that occur in it, is called contradiction.
Truth table of contradiction
๐ ~๐ ๐ ห~๐
T F F
F T F
19. Definition 11. Predicate
A predicate is a statement or mathematical
assertion that contains variables, sometimes
referred to as predicate variables, and may be true
or false depending on those variables values.
Example. ๐๐๐ก ๐(๐ฅ) denote the statement โ๐ฅ > 3โ
what are the truth values of ๐(4) and ๐(2) ?
Solution . By substituting x = 4,2
๐(4) is โ4> 3โ , which is true and
๐(2) is โ2> 3โ , which is false
20. Definition 12. Universal Quantification
The universal quantification of ๐(๐ฅ) is the
statement
โ ๐(๐ฅ) for all values of x in the domainโ.
The notation โ๐ฅ ๐(๐ฅ) denotes the universal
quantification of ๐(๐ฅ) . Here โ is called the
universal quantifier .An element for which ๐(๐ฅ) is
false is called a counterexample of โ๐ฅ ๐(๐ฅ)
21. Example1. Let ๐(๐ฅ), be the statement โ ๐ฅ + 1 > ๐ฅ. โ
what is the truth value of the quantification โ๐ฅ ๐(๐ฅ),
where the domain consists of all real numbers?
Solution : Because ๐ ๐ฅ is true for all real numbers x, the
quantification โ๐ฅ ๐ ๐ฅ is true.
Example 2.Let ๐ ๐ฅ be the statement โ๐ฅ < 2.โ what is the
truth value of the quantification โ๐ฅ๐ ๐ฅ , where the
domain consists of all real numbers?
Solution : ๐ ๐ฅ is not true for every real number ๐ฅ,
because for ๐ 3 is false i.e. 3 โฎ 2, x =
3 ๐๐ ๐๐๐ข๐๐ก๐๐ ๐๐ฅ๐๐๐๐๐ ๐๐๐ ๐กโ๐ ๐ ๐ก๐๐ก๐๐๐๐๐ก
thus โ๐ฅ๐ ๐ฅ is false
22. Definition 12. Existential Quantification
The existential quantification of ๐(๐ฅ) is the statement
โ There exists ๐๐ ๐๐๐๐๐๐๐ก ๐ฅ ๐๐ ๐กโ๐ ๐๐๐๐๐๐ ๐ ๐ข๐โ ๐กโ๐๐ก
๐(๐ฅ)โ. The notation โ๐ฅ ๐(๐ฅ) denotes the existential
quantification of ๐(๐ฅ) . Here โ is called the existential
quantifier .
Example .Let ๐ ๐ฅ be the statement โ๐ฅ = ๐ฅ + 1.โ what is
the truth value of the quantification โ ๐ฅ๐ ๐ฅ , where the
domain consists of all real numbers?
Solution : ๐ ๐ฅ is false for every real number ๐ฅ, the
existential quantification of ๐ ๐ฅ which is โ ๐ฅ๐ ๐ฅ ,is false