8.2 Symbolic Notation

Symbolic notation is used in logic and mathematics to express statements and relationships clearly and concisely.

Key Concepts:

• Logical Statements:

  A statement is a sentence that is either true or false but not both.

• Truth Values:

  A statement can be either: \[ \text{True (T)} \quad \text{or} \quad \text{False (F)} \]

• Logical Connectives:

  • **Negation (\(\neg P\)):** The opposite of a statement. \[ \text{If } P \text{ is "It is raining," then } \neg P \text{ is "It is not raining."} \]
  • **Conjunction (\(P \land Q\)):** True if both statements are true. \[ P \land Q \text{ is true only if both } P \text{ and } Q \text{ are true.} \]
  • **Disjunction (\(P \lor Q\)):** True if at least one statement is true. \[ P \lor Q \text{ is true if either } P \text{ or } Q \text{ (or both) are true.} \]
  • Contrapositive: The contrapositive of an implication states that if \(A \Rightarrow B\) is true, then its contrapositive \(\neg B \Rightarrow \neg A\) is also true. \[ (A \Rightarrow B) \Leftrightarrow (\neg B \Rightarrow \neg A) \]

    Example:

    If the statement "If it is raining, then the ground is wet" (\(A \Rightarrow B\)) is true, then the contrapositive "If the ground is not wet, then it is not raining" (\(\neg B \Rightarrow \neg A\)) must also be true.
  • **Implication (\(P \Rightarrow Q\)):** If \(P\) is true, then \(Q\) must be true. \[ \text{"If it rains, then the ground is wet."} \]
  • Contrapositive Rule: If an implication \(A \Rightarrow B\) is true, then its contrapositive \(\neg B \Rightarrow \neg A\) is also true. This means that if "A implies B" is valid, then "Not B implies Not A" is also valid. \[ (A \Rightarrow B) \Leftrightarrow (\neg B \Rightarrow \neg A) \]

    Example:

    If "If it is raining, then the ground is wet" (\(A \Rightarrow B\)) is true, then "If the ground is not wet, then it is not raining" (\(\neg B \Rightarrow \neg A\)) must also be true.
  • **Biconditional (\(P \Leftrightarrow Q\)):** True if \(P\) and \(Q\) are either both true or both false. \[ \text{"A shape is a square if and only if it has four equal sides and right angles."} \]

Examples:

Example 1: Determine the truth value of the following statement:

"If 2 is an even number, then 3 is an odd number."

Solution:

The statement can be written as: \[ P \Rightarrow Q, \quad \text{where } P: \text{"2 is even"} \text{ and } Q: \text{"3 is odd"}. \] Since both \(P\) and \(Q\) are true, the implication is **true**.

Example 2: Determine the truth value of the following compound statement:

"It is raining and it is sunny."

Solution:

The statement is of the form \(P \land Q\). If it is raining (\(P\) is true) but it is not sunny (\(Q\) is false), then: \[ P \land Q = F. \] Thus, the statement is **false**.

Example 2: Contrapositive of a Statement

Consider the statement: "If a number is divisible by 6, then it is divisible by 2."

Solution:

This can be written as an implication: \[ A \Rightarrow B, \quad \text{where } A: \text{"A number is divisible by 6"} \text{ and } B: \text{"The number is divisible by 2"}. \] The contrapositive of this statement is: \[ \neg B \Rightarrow \neg A, \quad \text{"If a number is not divisible by 2, then it is not divisible by 6."} \] Since this contrapositive is logically equivalent to the original statement, it must also be **true**.