Mathematical reasoning 

Statement

Statement: A sentence is called a mathematically acceptable if it is either true or false but not both.

The following are the Statements:

1) 6 is less than 9.  = True

2) 2 is an odd number. = False

3) Every square is a rectangle. = True

4) There are 10 Days in the week. = False

5) Gujrat is in India. = True

NOTE: 

A sentence which is both True and False simultaneously is not a statement. Such a sentence called paradox.

Open statement 

An open statement, also known as an open sentence or an open formula, is a statement that contains one or more variables or placeholders and becomes a statement when specific values are assigned to those variables. It is an expression that is not yet evaluated as true or false until the variables are given specific values.

Here are a few key points about open statements:

1. Variables: Open statements contain variables, which are symbols that represent unspecified values. The variables can be represented by letters or other symbols and can represent any element or quantity from a given set.

2. Examples:

   - "x + 3 = 7" is an open statement involving the variable x.

   - "n is an even number" is an open statement involving the variable n.

3. Truth Value: An open statement does not have a truth value until specific values are assigned to the variables. Once values are assigned, the open statement becomes a closed or specific statement, and its truth value can be determined.

4. Quantifiers: Open statements can be used with quantifiers to create statements that assert properties about a variable over a specific domain. For example:

   - "For all x, x + 2 > 5" is an open statement with a universal quantifier (∀) that asserts that for any value of x, x + 2 is greater than 5.

   - "There exists an x such that x^2 = 16" is an open statement with an existential quantifier (∃) that asserts that there exists at least one value of x such that x^2 is equal to 16.

Open statements are foundational in mathematical logic and are used in various branches of mathematics, including algebra, calculus, and set theory. They allow mathematicians to express relationships, properties, and equations in a more general and flexible manner before assigning specific values and making truth evaluations.

Compound statement 

A compound statement in mathematics is a statement that is formed by combining two or more simpler statements using logical connectives. Logical connectives include "and" (conjunction), "or" (disjunction), "if-then" (implication), and "if and only if" (biconditional). 

For example, let's consider two simple statements:

P: "The sun is shining."

Q: "It is raining."

We can combine these statements using logical connectives to form compound statements:

1. Conjunction (and): P and Q

   This compound statement would be "The sun is shining and it is raining."

   It would be true only if both P and Q are true.

2. Disjunction (or): P or Q

   This compound statement would be "The sun is shining or it is raining."

   It would be true if either P or Q or both are true.

3. Implication (if-then): P implies Q

   This compound statement would be "If the sun is shining, then it is raining."

   It would be true unless P is true and Q is false. In other words, it is false only if P is true and Q is false.

4. Biconditional (if and only if): P if and only if Q

   This compound statement would be "The sun is shining if and only if it is raining."

   It would be true if P and Q have the same truth value. In other words, it is true when both P and Q are true or both P and Q are false.

We will discuss these in more depth in following topics. 

Compound statements allow us to express more complex relationships and conditions, and they are important in mathematical reasoning and logic.

Simple statement

A simple statement, also known as an atomic statement or a basic statement, is a statement that cannot be further broken down into simpler statements. It is a statement that asserts a single fact or makes a single claim.

Simple statements typically represent basic facts or observations and are often used as building blocks for constructing more complex statements and logical arguments.

Here are some examples of simple statements:

1. "The sky is blue."

2. "2 + 2 = 4."

3. "The cat is on the mat."

4. "Today is Monday."

5. "Water boils at 100 degrees Celsius."

Each of these statements expresses a single proposition and does not contain logical connectives like "and," "or," or "if-then." Simple statements are considered the foundational elements of logical reasoning and form the basis for constructing more intricate mathematical and logical structures.

Component statement 

A component statement, also known as a compound component or elementary statement, is a statement that cannot be further divided into simpler statements. It is a basic building block in logical reasoning and serves as a component of more complex statements.

Component statements are typically used in logical expressions and logical operations. They are often combined with other component statements using logical connectives to form compound statements.

Here are some examples of component statements:

1. "P": The sun is shining.

2. "Q": It is raining.

3. "R": The train is delayed.

4. "S": The car is red.

5. "T": The number is even.

Each of these statements represents a single proposition and cannot be broken down into simpler statements. They can be used as variables or placeholders in logical expressions and combined using logical connectives such as "and," "or," "if-then," or "if and only if" to form more complex statements.

For instance, we can combine component statements P and Q using logical connectives:

- P and Q: The sun is shining and it is raining.

- P or Q: The sun is shining or it is raining.

- P implies Q: If the sun is shining, then it is raining.

- P if and only if Q: The sun is shining if and only if it is raining.

Component statements provide the basic units for constructing logical arguments, truth tables, and logical reasoning in mathematics and logic.

Negation

Negation is a logical operation that involves the denial or negation of a statement. It is a way to express the opposite or contradictory meaning of a given statement. The negation of a statement is often denoted by placing a negation symbol, such as "¬" or "~", in front of the statement.

For example, if we have a statement P: "The sun is shining," the negation of P would be denoted as ¬P or ~P and would express the opposite meaning, such as "The sun is not shining" or "It is not the case that the sun is shining."

Here are a few key points about negation:

1. Truth Value: The negation of a true statement is false, and the negation of a false statement is true. If a statement is neither true nor false (e.g., an open statement), its negation may be treated as the opposite assumption.

2. De Morgan's Laws: De Morgan's Laws provide rules for negating compound statements. According to these laws, the negation of a conjunction (and) is the disjunction (or) of the negations of the individual statements, and the negation of a disjunction (or) is the conjunction (and) of the negations of the individual statements.

• ¬(P and Q) is equivalent to (¬P or ¬Q)
• ¬(P or Q) is equivalent to (¬P and ¬Q)

3. Double Negation: Double negation states that the negation of a negation of a statement is equivalent to the original statement. In other words, ~~P is equivalent to P.

Negation is a powerful tool in logic and mathematics as it allows us to express the opposite of a given statement or proposition. It is widely used in constructing logical arguments, analyzing logical relationships, and forming complex statements.

Conjunction

Conjunction is a logical operation that combines two or more statements to form a compound statement. It is often represented by the word "and" and denoted by the symbol "∧". In a conjunction, all the component statements must be true for the compound statement to be true.

Here are some key points about conjunction:

1. Truth Table: The truth table for conjunction shows the possible combinations of truth values for the component statements and the resulting truth value of the conjunction. The truth table for a conjunction of two statements, P and Q, is as follows:

P	Q	P ∧ Q
T	T	T
T	F	F
F	T	F
F	F	F

2. Logical Operation: Conjunction combines two statements to create a compound statement that is true only when both component statements are true. For example, if we have the statements P: "The sun is shining" and Q: "It is a clear day," the conjunction P ∧ Q would be true if both the sun is shining and it is a clear day.

3. Associativity: Conjunction is associative, which means that when there are more than two component statements, the grouping does not affect the truth value of the compound statement. For example, (P ∧ Q) ∧ R is equivalent to P ∧ (Q ∧ R).

4. Precedence: In the order of logical operations, conjunction typically takes precedence over disjunction (or). So, if both conjunction and disjunction appear in the same compound statement without parentheses, conjunction is evaluated first.

NOTE: Do not think that a statement with "And" is always a Compound statement 

for Example:

A mixture of alcohol and water can be separated by chemical method. (this statement is not the compound statement)

Conjunction is widely used in mathematical logic, reasoning, and proof constructions. It allows us to combine multiple statements and establish relationships between them. By utilizing conjunction, mathematicians can create more complex statements and analyze logical connections between different components.

Disjunction

Disjunction is a logical operation that combines two or more statements to form a compound statement. It is often represented by the word "or" and denoted by the symbol "∨". In a disjunction, at least one of the component statements must be true for the compound statement to be true.

Here are some key points about disjunction:

1. Truth Table: The truth table for disjunction shows the possible combinations of truth values for the component statements and the resulting truth value of the disjunction. The truth table for a disjunction of two statements, P and Q, is as follows:

P	Q	P ∨ Q
T	T	T
T	F	T
F	T	T
F	F	F

2. Logical Operation: Disjunction combines two statements to create a compound statement that is true if at least one of the component statements is true. For example, if we have the statements P: "The sun is shining" and Q: "It is a clear day," the disjunction P ∨ Q would be true if either the sun is shining or it is a clear day.

3. Associativity: Disjunction is associative, which means that when there are more than two component statements, the grouping does not affect the truth value of the compound statement. For example, (P ∨ Q) ∨ R is equivalent to P ∨ (Q ∨ R).

4. Precedence: In the order of logical operations, disjunction typically takes lower precedence than conjunction (and). So, if both disjunction and conjunction appear in the same compound statement without parentheses, conjunction is evaluated first.

Disjunction is commonly used in mathematical logic, reasoning, and proof constructions. It allows us to combine multiple statements and establish relationships between them. By utilizing disjunction, mathematicians can create more complex statements and analyze logical connections between different components.

Inclusive OR

"Inclusive OR" is a term sometimes used to refer to the logical operation of disjunction (denoted by the symbol "∨" or by the word "or"). In the context of logic and mathematics, the term "inclusive OR" emphasizes that both component statements can be true in a disjunction, not necessarily exclusive to each other.

Inclusive OR means that if either or both of the component statements are true, the compound statement (disjunction) is considered true. It allows for the possibility of both component statements being true simultaneously.

For example, consider the following statements:

P: "The sun is shining."

Q: "It is raining."

Inclusive OR (disjunction) would be expressed as P ∨ Q, which can be interpreted as "The sun is shining or it is raining." In this case, if the sun is shining (P is true), or if it is raining (Q is true), or if both conditions are true, then the disjunction is true. It is inclusive in the sense that it includes the possibility of both statements being true.

The term "inclusive OR" is used to contrast with the concept of "exclusive OR" (XOR), where the disjunction is true only if exactly one of the component statements is true, excluding the case where both statements are true simultaneously.

It's worth noting that in most contexts, when the term "OR" is used without qualification, it typically refers to the inclusive OR unless otherwise specified.

Exclusive OR

Exclusive OR (XOR) is a logical operation that combines two statements to form a compound statement. It is often represented by the symbol "⊕" or by the phrase "either...or, but not both." In an exclusive OR, only one of the component statements can be true for the compound statement to be true.

Here are some key points about exclusive OR:

1. Truth Table: The truth table for exclusive OR shows the possible combinations of truth values for the component statements and the resulting truth value of the XOR operation. The truth table for XOR of two statements, P and Q, is as follows:

P	Q	P ⊕ Q
T	T	F
T	F	T
F	T	T
F	F	F

2. Logical Operation: Exclusive OR combines two statements to create a compound statement that is true if exactly one of the component statements is true. If both or neither of the component statements are true, the compound statement is false. For example, if we have the statements P: "The sun is shining" and Q: "It is raining," the exclusive OR P ⊕ Q would be true if either the sun is shining or it is raining, but not both.

3. Properties: XOR has the properties of being associative and commutative, meaning that the grouping and order of the component statements do not affect the truth value of the compound statement.

4. Relationship to Inclusive OR: Exclusive OR (XOR) is the opposite of inclusive OR. Inclusive OR is true when at least one of the component statements is true, allowing for the possibility of both being true. Exclusive OR is true when exactly one of the component statements is true, excluding the case where both are true.

Exclusive OR is used in various areas of logic, mathematics, and computer science. It is employed in error detection, cryptography, circuit design, and other fields where the distinction between "either...or, but not both" is crucial.

De-Morgan's Law

De Morgan's laws can be expressed as formulas that describe the relationships between negations, conjunctions, and disjunctions of statements. The formulas for De Morgan's laws are as follows:

1. De Morgan's First Law:

   ¬(P ∧ Q) ≡ (¬P ∨ ¬Q)

   This formula states that the negation of a conjunction is equivalent to the disjunction of the negations of the individual statements.

2. De Morgan's Second Law:

   ¬(P ∨ Q) ≡ (¬P ∧ ¬Q)

   This formula states that the negation of a disjunction is equivalent to the conjunction of the negations of the individual statements.

These formulas provide a concise representation of De Morgan's laws and allow for the transformation and simplification of logical expressions. By applying these laws, one can manipulate and rearrange statements involving negations, conjunctions, and disjunctions to create equivalent expressions. These formulas are fundamental tools in propositional logic and have various applications in mathematics, computer science, and other fields where logical reasoning is involved.

Contrapositive

The contrapositive is a logical operation that establishes an equivalent statement by reversing the order and negating the antecedent (the "if" part) and the consequent (the "then" part) of an implication. It is often used in logical reasoning and proofs to simplify or analyze implications.

To find the contrapositive of an implication statement "P → Q," you negate both P and Q, and then reverse their order. The resulting contrapositive statement is "¬Q → ¬P," which can be read as "if not Q, then not P."

Here are some key points about the contrapositive:

1. Logical Operation: The contrapositive statement maintains the same logical relationship as the original implication but reverses the order and negates the antecedent and consequent. If the original implication is true, its contrapositive is also true, and vice versa.

2. Truth Value: The truth value of an implication and its contrapositive are always the same. If the original implication is true, then its contrapositive is true, and if the original implication is false, then its contrapositive is false.

3. Example: Let's consider the implication "If it is raining, then the ground is wet." Its contrapositive would be "If the ground is not wet, then it is not raining." Both statements convey the same logical relationship.

4. Use in Proofs: The contrapositive is often employed in mathematical proofs as a technique to establish the validity of an implication. Instead of directly proving the original implication, one can prove its contrapositive, which might be easier or more straightforward. Since the contrapositive and the original implication are logically equivalent, the proof of the contrapositive implies the truth of the original statement.

The contrapositive is a valuable tool in logical reasoning and proof construction. By using the contrapositive, mathematicians and logicians can analyze and establish the validity of implications, simplify logical statements, and derive new logical relationships.

Converse

The converse of an implication is a related statement that exchanges the positions of the antecedent (the "if" part) and the consequent (the "then" part) of the original implication. In other words, the converse of an implication "P → Q" is "Q → P."

Here are some key points about the converse of an implication:

1. Logical Operation: The converse of an implication asserts the opposite relationship as the original implication. While the original implication states that if P is true, then Q is true, the converse states that if Q is true, then P is true.

2. Truth Value: The truth value of an implication and its converse may differ. Just because the original implication is true does not necessarily mean that its converse is true. The truth of the converse depends on the specific relationship between the antecedent and the consequent.

3. Example: Let's consider the implication "If it is raining, then the ground is wet." The converse of this statement would be "If the ground is wet, then it is raining." While the original implication is generally true, the converse is not necessarily true. The ground can be wet due to other reasons besides rain, such as watering or a spill.

4. Distinction from Implication: It's important to note that the converse is not logically equivalent to the original implication. They represent different relationships and may have different truth values. While the original implication establishes a specific conditional relationship, the converse explores the reverse conditional relationship.

The converse of an implication is a distinct statement that can be explored and analyzed separately from the original implication. It allows for the examination of the opposite relationship between the antecedent and the consequent. However, it is important to remember that the truth of an implication does not imply the truth of its converse, as the converse might not hold in all cases.