DISCRETE MATHEMATICS | IMPORTANCE AND SUPPORT TO INDUSTRIAL DEVELOPMENT
Discrete Mathematics is an important area of mathematics that deals with objects that
are countable or finite. It has numerous applications in computer science,
cryptography, logic, and other fields. Here are some reasons why learning
discrete mathematics is essential:
1. 1. Computer Science: Discrete mathematics is the foundation of
computer science. It provides the mathematical basis for understanding
algorithms, data structures, and computational complexity theory.
2. 2. Cryptography: Cryptography uses discrete mathematics extensively to provide secure communication channels. It helps to understand the principles of encryption and decryption, and the underlying mathematical concepts that make it possible to protect sensitive information.
3. 3. Logic: Discrete mathematics is the basis of mathematical logic,
which is the study of reasoning and inference. It helps to understand the
principles of propositional logic, predicate logic, and other logical systems.
4. 4. Combinatorics: Discrete mathematics is also important in combinatorics, which is the study of counting and arranging objects. It provides mathematical tools for understanding permutations, combinations, and other combinatorial structures.
5. 5. Graph Theory: Graph theory is a branch of discrete mathematics that deals with graphs, which are mathematical structures that represent relationships between objects. Graph theory has numerous applications in computer science, transportation, biology, and social networks.
LOGIC AND PROOFS
The foundations of logic
and proofs are fundamental concepts in mathematics and are used extensively in
many areas of science and engineering. Here are some key ideas:
1. 1. Propositions and Connectives:
A proposition is a statement that can be either true or false,
but not both. Connectives, such as "and", "or",
"not", and "if-then", are used to combine propositions to
form more complex statements.
2. 2. Logical Operators:
Logical operators, such as "and", "or",
"not", and "if-then", have well-defined meanings in
mathematics. For example, "p and q" is true only when both p and q
are true, while "p or q" is true when at least one of p or q is true.
3. 3. Truth Tables:
Truth tables are used to define the meaning of logical operators. A
truth table shows the truth values of a compound proposition for all possible
combinations of truth values of its component propositions.
4. 4. Quantifiers:
Quantifiers, such as "for all" and "there exists", are used to express the generality of a proposition. For example, "for all x, p(x)" means that the proposition p(x) is true for every possible value of x.
5. 5. Proofs:
Proofs are used to demonstrate that a proposition is true. Proof is a sequence of logical deductions that starts with axioms or previously proven propositions and ends with the desired result.
6. 6. Direct Proof:
A direct proof is a type of proof that starts with the given information and uses logical deductions to arrive at the desired conclusion.
7. 7. Proof by Contradiction:
A proof by contradiction is a type of proof that assumes the
opposite of the desired conclusion and shows that this leads to a
contradiction.
8. 8. Mathematical Induction:
Mathematical induction is a method of proof that is used to prove statements that have the form "for all n in the set of natural numbers, p(n) is true". The proof consists of two steps: the base case, where the statement is shown to be true for n=1, and the induction step, where the statement is shown to be true for n=k+1, assuming that it is true for n=k.
Let's have a few mathematical statements:
1. 1. 2 + 2 = 4
This is a
basic arithmetic statement that says the sum of 2 and 2 equals 4.
2. 2. x + 3 = 7
This is an algebraic statement that says an unknown quantity x added to 3 equals 7.
3. 3. 5 is a prime number.
This is a statement about the number 5 and its properties. It asserts that 5 is a prime number, which means it is only divisible by 1 and itself.
4. 4.. The area of a circle equals π times the radius squared.
This is a geometric statement that relates the area of a circle to its radius. It asserts that the area of a circle is equal to the constant π times the radius squared.
5. 5. The integral of f(x) with respect to x from 0 to 1 is equal to 3.
This is a calculus statement that relates a function f(x) to its integral. It asserts that the integral of f(x) with respect to x from 0 to 1 equals 3.
6. 6. If a and b are integers, then a + b is also an integer.
This is a statement about integers and their properties. It asserts that if a and b are integers, then their sum a + b is also an integer.
7. 7. For all real numbers x, x squared is greater than or equal to 0
This is a statement about the properties of real numbers. It asserts that the square of any real number is greater than or equal to 0.
Proposition and Logic
Propositional logic allows us to reason about the truth or falsity of complex statements based on the truth values of their component propositions. For example, suppose we have two propositions, p and q. The following are examples of compound propositions that can be formed using logical operators:
- p and q: This is true
only when both p and q are true.
- p or q: This is true
when at least one of p or q is true.
- not p: This is true
when p is false.
- p if and only if q:
This is true when either both p and q are true or both p and q are false.
Propositional logic also provides a way to analyze arguments for
their validity. An argument is valid if the truth of its premises implies the
truth of its conclusion. Propositional logic can be used to formalize arguments
in order to determine their validity. For example, consider the following
argument:
- If it rains today, I
will stay inside.
- It is raining today.
- Therefore, I will stay inside.
- p -> q (If it rains
today, I will stay inside)
- p (It is raining
today)
- Therefore, q (I will
stay inside)
By analyzing the truth values of the premises and the conclusion
using truth tables, we can see that the argument is valid since the truth of
the premises implies the truth of the conclusion.
Propositional logic is an important tool in
mathematics, computer science, and philosophy, and is the basis for more
advanced logical systems such as predicate logic and modal logic.
In logic, a proposition is a declarative statement that can
be either true or false. Propositions are the basic building blocks of logical
reasoning and form the basis of many advanced logical systems. Propositions can
be represented using variables, such as p and q, and can be combined using
logical operators, such as "and", "or", "not",
and "if-then". Here are some examples of propositions:
2. 2. "2 + 2 = 5." - This is a proposition because it is a statement that can be evaluated as either true or false. In this case, it is false.
3. 3 . "All dogs have tails." - This is a proposition because it is a statement that can be evaluated as true or false. In this case, it is true because all dogs do indeed have tails.
4.
4. "The Earth is flat." - This is a proposition because
it is a statement that can be either true or false. In this case, it is false
as scientific evidence has proven the Earth is a sphere.
5. 5. "John is taller than Peter." - This is a proposition because it is a statement that can be evaluated as true or false. In this case, it is either true or false depending on the height of John and Peter.
Proposition using p and q in a statement
1. 1. p and q - This is a proposition that
can be evaluated as true or false depending on the truth values of p and q. It
is true only when both p and q are true.
2. 2. p
or q - This is a proposition that can be evaluated as true or false depending
on the truth values of p and q. It is true when at least one of p or q is true.
3. 3. not
p - This is a proposition that can be evaluated as true or false depending on
the truth value of p. It is true when p is false.
4. 4. p
-> q - This is a proposition that states "if p, then q". It is
true unless p is true and q is false.
5.
5. p if and only if q - This is a proposition that states "p is true if and only if q is true". It is true if either both p and q are true or both p and q are false.
For example, let's assume p is "It is raining today" and q is "I will bring an umbrella". Then, we can give an example for each of the above propositions:
1. 1. "It
is raining today and I will bring an umbrella." - This proposition is true
only if both p and q are true.
2. 2. "It is raining today or I will bring an umbrella." - This proposition is true if at least one of p or q is true.
3. 3. "It is not raining today." - This proposition is true when p is false.
4. 4. "If it is raining today, then I will bring an umbrella." - This proposition is true unless it is raining today and I don't bring an umbrella.
5. 5. "I will bring an umbrella if and only if it is raining today." - This proposition is true if either both it is raining today and I bring an umbrella, or it is not raining today and I don't bring an umbrella.
PROPOSITIONAL LOGIC
In propositional logic, proposition symbols are symbols that represent propositions or statements that are either true or false. These symbols are typically represented by capital letters such as P, Q, R, etc.
Proposition symbols are used in propositional logic to construct
compound statements by combining them with logical operators such as AND (∧), OR (∨), NOT (¬), IMPLIES (→), and EQUIVALENT (↔).
For example, if we let P represent the statement "It is
raining" and Q represent the statement "I am indoors", we can
use the logical operator AND (∧)
to construct the compound statement "It is raining AND I am indoors",
which would be represented as P ∧ Q. Similarly, the statement "It is not
raining" could be represented as ¬P.
Proposition symbols are a fundamental concept
in propositional logic and are used extensively in many fields, including
computer science, mathematics, and philosophy.
EXAMPLES OF COMPOUND PROPOSITIONAL LOGIC USING LOGICAL SYMBOLS.
Let p - The sky is blue
q - it is
daytime
1.
(p ∧ q): The sky is
blue AND it is daytime.
Explanation: This
compound proposition represents the conjunction of two simpler propositions: P,
which represents the proposition "The sky is blue", and Q, which
represents the proposition "It is daytime". The entire compound
proposition is true only if both P and Q are true.
2.
(p ∨ q): The sky is
blue OR it is daytime.
Explanation: This
compound proposition represents the disjunction of two simpler propositions: R,
which represents the proposition "All cats are mammals", and S, which
represents the proposition "The capital of Italy is Rome". The entire
compound proposition is true if either R or S (or both) are true.
3.
(¬P): The sky is blue
Explanation: This
compound proposition represents the negation of a simpler proposition: P, which
represents the proposition "The sun is shining". The entire compound
proposition is true if P is false.
4.
(p → q): If The sky
is blue then it is daytime.
Explanation: This compound proposition represents a conditional
statement, where Q represents the antecedent or premise "It is
raining", and R represents the consequent or conclusion "The plants
will grow". The entire compound proposition is true if either Q is false
or R is true.
5.
(p ↔ q): The sky is blue if and only if it
is daytime.
Explanation: This compound proposition represents a
biconditional statement, where P represents the proposition "The
temperature is hot", and Q represents the proposition "The air
conditioner is on". The entire compound proposition is true only if both P
and Q are true, or if both P and Q are false.
1. 1. P: It is raining outside.
Answer: True or False, depending on
whether it is actually raining outside.
2. 2. Q: 5 + 7 = 12.
Answer: False, because 5 + 7 equals 12 is a false statement.
3. 3. R: All dogs are mammals.
Answer: True, because it is a true statement that all dogs are mammals.
4. 4. S: The moon is made of green cheese.
Answer: False, because the moon is not made of green cheese.
5. 5. T: If it is Monday, then it is a workday.
Answer: True or False, depending on whether it is Monday and whether it is a workday.
1. P: It is raining
outside.
Q: I have an umbrella.
Statement: If it is raining outside and I have an umbrella, then I will go for a walk.
Answer: True or False, depending on the truth values of P
and Q.
2. R: I have a car.
S: I have enough money.
Statement: If I have a car or enough money, then I can go on a
road trip.
Answer: True, as long as either R or S is true.
3. T: I will study hard.
U: I will pass the exam.
Statement: I will pass the exam if and only if I study hard.
Answer: True, if T is true and U is true; False, if T is false
and U is false; otherwise, the statement is False.
4. V: It is hot outside.
W: I am thirsty.
Statement: If it is hot outside, then I am thirsty.
Answer: The statement is True if V is true and W is true or
false; the statement is False, only if V is false and W is true.
5. X: I will wake up early.
Y: I will exercise.
Z: I will have a healthy breakfast.
Statement: If I wake up early, then I will exercise and have a healthy breakfast.
Answer: The statement is True if X is true and Y is true and Z is true; False, if X is false and Y is false or Z is false.
Compound and non Compound Logical Statements
1. 1. It is sunny today. (Non-compound)
2. 2. If it rains, then I will stay indoors.
(Compound)
3. 3. I will buy a new car or repair the old one.
(Compound)
4. 4. All cats are mammals. (Non-compound)
5. 5. If I study hard, then I will get good grades.
(Compound)
6. 6. John is taller than Tom but shorter than
Mike. (Compound)
7. 7. It is not raining outside. (Non-compound)
8. 8. I will go to the gym if I have time and
energy. (Compound)
9. 9. The Earth revolves around the Sun.
(Non-compound)
10. If it is not snowing, then I will go skiing. (Compound)
REVIEW QUESTIONS WITH ANSWERS
1. What is the truth value of the proposition "P ∧ ~P"?
A. True
B. False
C. Cannot be determined
Answer:: B. False
2. Which of the following is the negation of the proposition "P ∨ Q"?
A. ~P ∧ ~Q
B. ~P ∨ ~Q
C. P ∧ Q
Answer: A. ~P ∧ ~Q
3. What is the truth value of the proposition "P → Q" when P is false and Q is true?
A. True
B. False
C. Cannot be determined
Answer: A. True
4. What is the contra-positive of the proposition "P → Q"?
A. ~Q → ~P
B. P → ~Q
C. ~P → Q
Answer:: A. ~Q → ~P
5. Which of the following is equivalent to the proposition "P ↔ Q"?
A. (P → Q) ∧ (Q → P)
B. ~(P ∧ Q)
C. (P ∨ Q) → ~(P ∧ Q)
Answer: A. (P → Q) ∧ (Q → P)
6. What is the truth value of the proposition "P ∨ (Q ∧ ~Q)"?
A. True
B. False
C. Cannot be determined
Answer:: A. True
7. What is the inverse of the proposition "P → Q"?
A. P ∧ ~Q
B. ~P → ~Q
C. ~P ∧ Q
Answer: C. ~P ∧ Q
8. What is the truth value of the proposition "~(P ∧ Q)" when P is false and Q is true?
A. True
B. False
C. Cannot be determined
Answer: A. True
9. What is the converse of the proposition "P → Q"?
A. Q → P
B. ~Q → ~P
C. ~P → Q
Answer:: A. Q → P
10. Which of the following is equivalent to the proposition "(P ∧ Q) → R"?
A. (P → R) ∧ (Q → R)
B. (P → Q) ∨ (Q → P)
C. ~(P ∧ Q) ∨ R
Answer: A. (P → R) ∧ (Q → R)
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