Mathematics in the Modern World: Language and Logic

Conceptual Framework of Mathematics

• Mathematics is multidimensional in its definition and application within the modern world. It can be characterized as follows:

  • A Study of Patterns: Investigating regularities and structures in both the natural and abstract worlds.

  • A Language: A systematic method of communicating complex ideas through specialized symbols and syntax.

  • An Art: A creative pursuit involving aesthetics, symmetry, and elegance in reasoning.

  • A Set of Problem Solving Tools: A practical toolkit used to analyze, quantify, and solve real-world challenges.

  • A Process of Critical Thinking: A rigorous method of logical reasoning and analytical thought.

The Nature of Mathematical Language

• Definition of Language: Language is defined as a systematic way of communicating with other people through the use of sounds or conventional symbols.

• Mathematical language specifically utilizes symbols and formal structures to ensure precision and remove the ambiguity often found in natural languages.

Statements and Propositions in Logic

• Definition of a Statement (Proposition): A statement or proposition is a declarative sentence that is true or false, but not both.

• Types of Statements:

  • Simple Statements: A statement that conveys a single idea or assertion.

  • Compound Statements: A statement that conveys two or more ideas, usually linked by logical connectives.

Analysis of Propositions: Examples and Validations

• Example 1: "Tagum City is the capital of Davao del Norte."

  • Classification: Proposition.

  • Reasoning: It is a declarative sentence that has a truth value (in this case, TRUE).

• Example 2: "Pay attention to this."

  • Classification: Not a Proposition.

  • Reasoning: The sentence is imperative (a command), not declarative; therefore, it cannot be assigned a truth value of true or false.

• Example 3: $1 + 1 = 2$

  • Classification: Proposition.

  • Reasoning: It is a mathematical sentence with a truth value (TRUE).

• Example 4: $x + y = z$

  • Classification: Not a Proposition.

  • Reasoning: The sentence is neither true nor false because the values of the variables are not specified (this is known as an open sentence).

• Example 5: x > 5 if $x = 3$

  • Classification: Proposition.

  • Reasoning: Because a specific value is assigned to the variable, the sentence has a definite truth value (in this case, FALSE, as $3$ is not greater than $5$).

Logical Connectives and Symbolic Notation

• Logical connectives are used to join simple propositions into compound propositions. The primary connectives include:

• Conjunction:

  • Symbol: $\land$

  • English Keyword: "and"

• Disjunction:

  • Symbol: $\lor$

  • English Keyword: "or"

• Conditional:

  • Symbol: $\rightarrow$

  • English Keyword: "if … then"

• Biconditional:

  • Symbol: $\leftrightarrow$

  • English Keyword: "if and only if"

• Negation:

  • Symbol: $\neg$

  • English Keyword: "not"

Application of Compound Propositions

• Let $p$ and $q$ be defined as the following propositions:

  • $p$: Today is Friday.

  • $q$: It is raining today.

• Conjunction ($p \land q$): "Today is Friday and it is raining today."

• Disjunction ($p \lor q$): "Today is Friday or it is raining today."

• Conditional ($p \rightarrow q$): "If today is Friday then it is raining today."

• Biconditional ($p \leftrightarrow q$): "Today is Friday if and only if it is raining today."

• Negation ($\neg p$): "Today is not Friday."

• Negation ($\neg q$): "It is not raining today."

Exercises in Propositional Logic

• Evaluating Sentences (Determine if Proposition or Not):

  1. "Manila is the capital of the Philippines."

  2. $2 + 2 = 3$

  3. "Come to class!"

  4. "$n$ is a prime number."

  5. -1 < 0

• Translating Symbolic Logic to English:

  • Using the propositions $p$ (Today is Friday) and $q$ (It is raining today), express the following:

  1. $p \land \neg q$: Today is Friday and it is not raining today.

  2. $\neg p \lor q$: Today is not Friday or it is raining today.

  3. $\neg p \rightarrow \neg q$: If today is not Friday, then it is not raining today.

  4. $\neg p \leftrightarrow \neg q$: Today is not Friday if and only if it is not raining today.