Conditional Statements: if p then q

Introduction

This tutorial focuses on understanding conditional statements, particularly the "if p then q" format. By the end, you will be able to interpret conditional statements, construct truth tables, and recognize logical equivalences in different forms. This knowledge is essential in discrete mathematics and helps in logical reasoning.

Step 1: Interpret Conditional Statements

• A conditional statement is typically expressed as "if p then q," where:
  • p is the hypothesis (the condition).
  • q is the conclusion (the result).
• Example interpretation:
  • If it rains (p), then the ground will be wet (q).
• Key points to remember:
  • The truth of the statement depends on the relationship between p and q.
  • A conditional statement can be true or false based on the truth values of p and q.

Step 2: Write the Truth Table for a Conditional

• A truth table displays all possible truth values for p and q, and the outcome of "if p then q."
• Here’s how to construct it:
  1. List all combinations of truth values for p and q:
     • True (T) and True (T)
     • True (T) and False (F)
     • False (F) and True (T)
     • False (F) and False (F)
  2. Determine the truth value of "if p then q" for each combination:
     • T → T = T
     • T → F = F
     • F → T = T
     • F → F = T
• The completed truth table looks like this:

| p | q | p → q | |-------|-------|-------| | T | T | T | | T | F | F | | F | T | T | | F | F | T |

Step 3: Understand Logical Equivalence

• A conditional statement can be expressed in different forms, and it's important to recognize their equivalence.
• The disjunctive form of "if p then q" is logically equivalent to "not p or q" (¬p ∨ q).
• To verify this, you can use the truth table:
  • Create a truth table for ¬p ∨ q and compare it with the truth table for p → q.
• The truth table for ¬p ∨ q will match that of p → q, confirming their equivalence.

| p | q | ¬p | ¬p ∨ q | |-------|-------|-------|--------| | T | T | F | T | | T | F | F | F | | F | T | T | T | | F | F | T | T |

Conclusion

In this tutorial, you learned how to interpret conditional statements, construct truth tables, and understand the logical equivalence of different forms of conditional statements. To solidify your understanding, practice creating your own truth tables and exploring various examples of conditional statements. This foundational knowledge will enhance your skills in logical reasoning and discrete mathematics.