Fundamentals of Mathematical Logic

Sure, let’s dive into the world of mathematical logic! Mathematical logic is a branch of mathematics that explores the principles of reasoning and deduction using formal mathematical methods. It’s used in various fields like computer science, philosophy, linguistics, and mathematics itself. Here, we’ll discuss some solved exercises in mathematical logic to give you a better understanding.

1. Propositional Logic:
   Let’s start with basic propositions and logical operations. Consider the following propositions:

   • $P$: “It is sunny today.”
   • $Q$: “I will go for a walk.”
   • $R$: “I will have ice cream.”

   Now, let’s express some logical statements and solve them:

   • If it is sunny today ($P$), then I will go for a walk ($Q$).
     • $P \rightarrow Q$
   • I will not have ice cream ($\neg R$) if I don’t go for a walk ($\neg Q$).
     • $\neg Q \rightarrow \neg R$
   • If it is not sunny today ($\neg P$), then I will not go for a walk ($\neg Q$).
     • $\neg P \rightarrow \neg Q$

2. Predicate Logic:
   Next, let’s move to predicate logic, which deals with predicates and quantifiers. Consider the following predicates:

   • $P(x)$: “x is an even number.”
   • $Q(x)$: “x is greater than 10.”
   • $R(x, y)$: “x is the parent of y.”

   Now, let’s express some logical statements and solve them:

   • All even numbers are greater than 10.
     • $\forall x (P(x) \rightarrow Q(x))$
   • No even number is the parent of any other number.
     • $\neg \exists x \exists y (P(x) \land R(x, y))$
   • There exists an even number that is not greater than 10.
     • $\exists x (P(x) \land \neg Q(x))$

3. Boolean Algebra:
   In Boolean algebra, we deal with logical operations on binary variables (0 and 1). Consider the following Boolean expressions:

   • $A$: 0
   • $B$: 1
   • $C$: 0

   Now, let’s solve some Boolean expressions:

   • $A + B \cdot C$
     • $0 + 1 \cdot 0 = 0$
   • $(A + B) \cdot (A + C)$
     • $(0 + 1) \cdot (0 + 0) = 1 \cdot 0 = 0$
   • $A \cdot (B + C)$
     • $0 \cdot (1 + 0) = 0$

4. Logical Equivalences:
   Logical equivalences are statements that are always true, regardless of the truth values of their components. Let’s explore some logical equivalences:

   • Double Negation: $\neg (\neg P) \equiv P$
   • De Morgan’s Laws: $\neg (P \land Q) \equiv \neg P \lor \neg Q$ and $\neg (P \lor Q) \equiv \neg P \land \neg Q$
   • Idempotent Laws: $P \lor P \equiv P$ and $P \land P \equiv P$
   • Absorption Laws: $P \lor (P \land Q) \equiv P$ and $P \land (P \lor Q) \equiv P$

5. Proof Techniques:
   In mathematical logic, proof techniques are used to establish the validity of logical statements. Here are some common proof techniques:

   • Direct Proof: Assume the premise and show step-by-step how it logically leads to the conclusion.
   • Proof by Contradiction: Assume the opposite of what you want to prove, derive a contradiction, and conclude that the original statement must be true.
   • Proof by Contrapositive: Instead of proving $P \rightarrow Q$, prove $\neg Q \rightarrow \neg P$, which is logically equivalent.
   • Proof by Mathematical Induction: Used to prove statements about natural numbers by showing a base case and then proving the inductive step.

These are just a few examples of exercises and concepts in mathematical logic. Each topic within mathematical logic has its own set of rules, operations, and techniques for solving problems and proving statements. If you’d like to delve deeper into any specific area or have more exercises, feel free to ask!

Certainly! Let’s delve deeper into the various aspects of mathematical logic and provide more comprehensive information.

1. Propositional Logic:

Propositional logic deals with propositions, which are statements that can be either true or false. In this branch of logic, we use logical connectives to form compound propositions and analyze their truth values.

Logical Connectives:

• Conjunction (AND, $\land$): Represents the logical “and” operation. $P \land Q$ is true only when both $P$ and $Q$ are true.
• Disjunction (OR, $\lor$): Represents the logical “or” operation. $P \lor Q$ is true if at least one of $P$ or $Q$ is true.
• Negation (NOT, $\neg$): Represents the logical negation. $\neg P$ is true if $P$ is false, and vice versa.
• Implication (IF…THEN, $\rightarrow$): Represents the logical implication. $P \rightarrow Q$ is false only when $P$ is true and $Q$ is false.
• Biconditional (IF AND ONLY IF, $\leftrightarrow$): Represents the logical equivalence. $P \leftrightarrow Q$ is true when $P$ and $Q$ have the same truth value.

Truth Tables: Truth tables are used to analyze the truth values of compound propositions based on the truth values of their components. For example:

• $P \land Q$:

Logical Equivalences:

• Associative Laws: $P \land (Q \land R) \equiv (P \land Q) \land R$ and $P \lor (Q \lor R) \equiv (P \lor Q) \lor R$
• Distributive Laws: $P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R)$ and $P \lor (Q \land R) \equiv (P \lor Q) \land (P \lor R)$
• Idempotent Laws: $P \land P \equiv P$ and $P \lor P \equiv P$
• De Morgan’s Laws: $\neg (P \land Q) \equiv \neg P \lor \neg Q$ and $\neg (P \lor Q) \equiv \neg P \land \neg Q$

2. Predicate Logic:

Predicate logic extends propositional logic by introducing predicates, variables, quantifiers, and functions. It allows us to make statements about classes of objects rather than specific instances.

Quantifiers:

• Universal Quantifier ($\forall$): Represents “for all” or “for every.” $\forall x P(x)$ means $P(x)$ is true for all values of $x$ in the domain.
• Existential Quantifier ($\exists$): Represents “there exists.” $\exists x P(x)$ means there exists at least one $x$ such that $P(x)$ is true.

Logical Statements:

• Implication with Quantifiers: $\forall x (P(x) \rightarrow Q(x))$ means “For all $x$, if $P(x)$ then (Q(x)).”
• Negation with Quantifiers: $\neg \exists x P(x)$ means “It is not true that there exists an $x$ such that $P(x)$.”
• Universal Generalization: If $P(c)$ is true for an arbitrary element $c$ from the domain, then $\forall x P(x)$ is true.

3. Mathematical Induction:

Mathematical induction is a proof technique used to prove statements about natural numbers. It consists of two steps: the base case and the inductive step.

Base Case:

• Prove that the statement holds true for a specific starting value, usually $n = 0$ or $n = 1$.

Inductive Step:

• Assume the statement is true for $n = k$.
• Prove that if it holds for $n = k$, then it must also hold for $n = k + 1$.

Example:
Let’s prove the sum of the first $n$ natural numbers using mathematical induction:

• Base Case: $n = 1$, $1 = \frac{1(1 + 1)}{2}$ is true.
• Inductive Step: Assume true for $n = k$, i.e., $1 + 2 + \dots + k = \frac{k(k + 1)}{2}$.
  • Prove for $n = k + 1$: $1 + 2 + \dots + k + (k + 1) = \frac{(k + 1)(k + 2)}{2}$.

4. Formal Proofs:

In formal logic, proofs are constructed using strict rules of inference. These rules ensure that the conclusions drawn are valid based on the premises.

Types of Proofs:

• Direct Proof: Start with the premises and use valid logical steps to reach the conclusion.
• Proof by Contradiction: Assume the opposite of what you want to prove and show that it leads to a contradiction, thus proving the original statement.
• Proof by Contrapositive: Prove the contrapositive of the statement, which is logically equivalent.
• Proof by Cases: Break down the problem into different cases and prove each case individually.

Example:
Prove that if $n$ is an odd integer, then $n^2$ is also odd.

• Direct Proof:
  • Let $n = 2k + 1$ for some integer $k$.
  • $n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1$, which is of the form $2m + 1$, proving that $n^2$ is odd.

These are fundamental concepts and techniques in mathematical logic. They form the basis for more advanced topics like set theory, formal languages, computability, and complexity theory. If you have specific topics or exercises you’d like to explore further, feel free to ask!