3. What Is Logic?
Logical Operations
Examples Of Logics
What Is Proofs?
Types Of Proofs
Example Of Proofs
Application Of Logic And Proofs With Example
CONTENTS OF THIS TEMPLATE
5. What Is Logic?
Logic means reasoning. The reasoning may be a
legal opinion or mathematical confirmation.
We apply certain logic in Mathematics. Basic
Mathematical logics are a negation,
conjunction, and disjunction. The symbolic
form of mathematical logic is, ‘~’ for
negation ‘^’ for conjunction and ‘ v ‘ for
disjunction
6. Logic Operations
We can join two
statements by
“AND” operand. It is
also known as a
conjunction. Its
symbolic form is “∧“.
In this operator, if
anyone of the
statement is false,
then the result will
be false. If both the
statements are true,
then the result will
be true. It has two or
more inputs but only
one output.
Conjuction(AND) Truth Table For And(^)
Input 1 Input 2 Output
A B A^B
T T T
T F F
F T F
F F F
7. Logic Operations
We can join two
statements by “OR”
operand. It is also
known as
disjunction. It’s
symbolic form is “∨”.
In this operator, if
anyone of the
statement is true,
then the result is
true. If both the
statements are false,
then the result will
be false. It has two
or more inputs but
only one output
Disjunction(OR) Truth Table For Or(V)
Input 1 Input 2 Output
A B AVB
T T T
T F T
F T T
F F F
8. Logical Operations
Negation is an
operator which gives
the opposite
statement of the
given statement. It is
also known as NOT,
denoted by “∼”. It is
an operation that
gives the opposite
result. If the input is
true, then the output
will be false. If the
input is false, then
the output will be
true. It has one input
and one output
Negation(NOT) Truth Table For Not(∼)
Input 1 Output
A NEGATION(~A)
T F
F T
9. Example Of Logic(AND)
Write the truth table values of conjunction for the given two statements
A: x is an even number
B: x is a prime number
Solution:
Given: A: x is an even number
B: x is a prime number
Let assume the different x values to prove the conjunction truth table
X Value A B (A^B)
X=2 T T T
X=4 T F F
X=3 F T F
X=9 F F F
10. Example Of Logic(OR)
Write the truth table values of disjunction for the given two statements
A: p is divisible by 2
B: p is divisible by 3
Solution:
Given: A: P is divisible by 2
B: P is divisible by 3
Let assume the different x values to prove the disjunction truth table
Value Of P A B (AVB)
P=12 T T T
P=4 T F T
P=9 F T T
P=7 F F F
11. Example Of Logic(NOT)
Find the negation of the given statement “ a number 6 is an even number”
Solution:
Let “S” be the given statement
S = 6 is an even number
Therefore, the negation of the given statement is
∼S = 6 is not an even number.
Therefore, the negation of the statement is “ 6 is not an even number”
13. What Is Proofs?
A proof is a logical argument that tries to show
that a statement is true. In math, and
computer science, a proof has to be well
thought out and tested before being
accepted. But even then, a proof can
be discovered to have been wrong. There are
many different ways to go about proving
something, we’ll discuss 3 methods: direct
proof, proof by contradiction, proof by
induction.
14. Types Of Proofs
Prove that if m+n and n+p are even integers where m, sp are integers, then m+p is even.
Solution: Given: If m+n is even and n+p is even then m+p
is even.
P: m+n is even and n + p is even.
Q: m+p is even.
1. Hypothesis: Assume that P is true.
2 Analysis: If m+n is even then m + n = 2s, for some integer 's If n + p is even then n+p=2t, for
some integer 't'.
m+p = (m + n) + (n+p) - 2n
=2s +2t-2n
=2(s+t-n)
3. Conclusion: We observe that R.H.S. value of m+p is divisible
This means that m+p is an even integer.
ie, P->Q is true.
Direct Proofs
Direct proofs lead from the hypothesis of a theorem to son. In a direct
proof, we assume that P is true and use anions dditions, and previous
theorems, together with rules of inference, to show that must also be true.
Examples
15. Types Of Proofs
Example 1. Prove that if n is an integer and 5n+2 is odd, then n is odd.
Solution: P:n is an integer and 5+2 is odd.
Q:n is odd.
To prove p→ q is true it is enough to prove ~q->~ P is true.
1. Hypothesis:
Since P->Q its contrapositive.
~q-~p are logically equivalent.
So assume that ~q is true.
ie., n is even.
2 Analysis: If n is even, then n = 2k, for some integer k.
5n+2=5 (2k) +2
= 2 (5k) + 2
= 2 (5k+1)
3. Conclusion:
We observe that R.HS value of 5n+ 2 is divisible by 2.,This means that Sn +2 is an even
integer.
ie, P is true.
~q->~ p is true.
Proof By
Contrapostion
A proof by contraposition is sometimes called an indirect proof. It will
make use of the fact that the conditional statement P→ Q can he proved by
showing that its contrapositive, ~q->~p is true
Examples
16. ● 1) Translating English Sentences into logical statements
● Like any other human language, English sentences can be
ambiguous. This ambiguity might lead to uninformed
decision-making and other fatal errors. To remove this
ambiguity, we can translate these English sentences into
logical expressions with the help of Propositional Logic. Note
that sometimes this may include making a few assumptions
based on the sentence’s intended meaning.
● Example: Given a sentence “You can purchase this book if
you have $20 or $10 and a discount coupon.” Now, this is a
bit complex to be understood at once. So we translate this
into a logical expression that will make it simple to
understand. Let a, b, c, and d represent the sentences “You
can purchase this book.“, “You have $20.“, “You have $10.“,
and “You have a discount coupon.” respectively. Then the
given sentence can be translated to (b ∨ (c ∧ d) -> a, which
simply means that “if you either have $20 or $10, along with
a discount coupon, then you can purchase the book.
Applications in Logic and Proof
17. ● Boolean Searches
● Another important application of propositional logic
is Boolean searches. These searches use techniques from
propositional logic. Logical connectives are used extensively
in searches of large collections of information, such as
indexes of Web Pages. In Boolean searches, the connective
AND is used to find records that contain both of the two
terms, the connective OR is used to find records that have
either one or both of the terms, and the connective NOT,
also written as AND NOT, is used when we need to exclude
a particular search term.
● Example: Some web search engines support web page
searching that uses Boolean techniques. For instance, if we
want to look for web pages about hiking in West Alps, then
we can look for pages matching
WEST AND ALPS AND HIKING. And if we want web pages
about hiking in the Alps, but not in West Alps, then we can
search for pages matching the record
HIKING AND ALPS AND NOT(WEST AND ALPS).
Applications in Logic and Proof
18. CREDITS: This presentation template was created by
Slidesgo, including icons by Flaticon, and
infographics & images by Freepik
THANKS
DONE BY:
BHARATH G
