2.4 Truth Tables for the Conditional and Biconditional

Conditional statements are like if-then promises in logic. They're only false when the "if" part is true but the "then" part isn't. This helps us understand cause-and-effect relationships in everyday life and math.

Biconditional statements are two-way streets, true when both parts match. They're useful for defining things precisely. Understanding these logical structures helps us reason better and solve problems more effectively.

Truth Tables for Conditional and Biconditional Statements

Truth tables for conditional statements

• Conditional statement has the form "if p, then q" or "p implies q", denoted as $p \rightarrow q$
  • $p$ represents the hypothesis (antecedent) and $q$ represents the conclusion (consequent)
  • Example: "If it rains (p), then the ground will be wet (q)"
• Truth table for a conditional statement $p \rightarrow q$ shows the statement is only false when the hypothesis is true and the conclusion is false
  • If the hypothesis is false, the conditional statement is always true, regardless of the truth value of the conclusion
  • Truth table:

• Real-world example: "If you study hard (p), then you will pass the exam (q)"
  • If you study hard and pass the exam, the statement is true
  • If you study hard but don't pass the exam, the statement is false
  • If you don't study hard, the statement is true regardless of whether you pass or fail the exam
• The contrapositive of a conditional statement is logically equivalent to the original statement

Validity of biconditional statements

• Biconditional statement has the form "p if and only if q", denoted as $p \leftrightarrow q$
  • Combination of a conditional statement and its converse: $(p \rightarrow q) \land (q \rightarrow p)$
  • Example: "A figure is a square (p) if and only if it has four equal sides and four right angles (q)"
• Biconditional statement is true when both $p$ and $q$ have the same truth value (both true or both false)
  • Truth table:

• Real-world example: "Two triangles are congruent (p) if and only if their corresponding sides and angles are equal (q)"
  • If the triangles are congruent and their sides and angles are equal, the statement is true
  • If the triangles are not congruent and their sides and angles are not equal, the statement is true
  • If the triangles are congruent but their sides and angles are not equal (or vice versa), the statement is false

Applications of conditional logic

• Conditional statements used in computer programming for if-else statements and loops
  • Example: "If the user enters a valid password (p), then grant access to the system (q)"
  • If the password is valid and access is granted, the statement is true
  • If the password is valid but access is not granted, the statement is false
  • If the password is invalid, the statement is true regardless of whether access is granted or not
• Biconditional statements used to define equivalence relations or check for equality between two conditions
  • Example: "A number is even (p) if and only if it is divisible by 2 (q)"
  • If a number is even and divisible by 2, the statement is true
  • If a number is not even and not divisible by 2, the statement is true
  • If a number is even but not divisible by 2 (or vice versa), the statement is false

Related Concepts

• Logical equivalence: Two statements are logically equivalent if they have the same truth values for all possible combinations of their component propositions
• Tautology: A compound statement that is always true, regardless of the truth values of its component propositions
• Converse and inverse: The converse of "if p, then q" is "if q, then p", while the inverse is "if not p, then not q"