Lecture 1: Introduction
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What Is a Proposition?
Definition
A proposition is a declarative sentence that is either True or False, but not both.
To qualify as a proposition:
It must have a definite truth value
It cannot depend on an undefined variable
It cannot be a command, question, or opinion
Examples (You MUST be able to do these)
Statement | Proposition? | Why |
|---|---|---|
"7 is prime." | ✅ Yes | Has a definite truth value (True) |
"2 is odd." | ✅ Yes | Has a definite truth value (False) |
"Please do your homework." | ❌ No | Command |
"What is your favorite color?" | ❌ No | Question |
"n + 1 is even." | ❌ No | Depends on undefined variable |
"If pigs fly, then I am rich." | ✅ Yes | Logical implication with truth value |
Exam Trap: Variables
If a statement depends on a variable without specifying it, it is NOT a proposition.
“n is even” → ❌ Not a proposition
“For all integers n, n is even” → ✅ Proposition (False)
Logical Implications & Vacuous Truth
Implication Form
An implication has the form:
Read as:
“If P is true, then Q is true.”
Truth Rules (Critical for Exams)
P | Q | |
|---|---|---|
T | T | T |
T | F | ❌ F |
.. | ... | ... |
... | ... | ... |
Vacu....
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