Lesson: Basic Logic
Hello everyone! Welcome to our lesson on Basic Logic, which is a core part of the Applied Mathematics 2 syllabus under the topic of Numbers and Algebra.
If you’ve ever felt that math is just full of headache-inducing numbers, this chapter will change your mind! Logic is all about "reasoning," which we use in our daily lives all the time—like saying "If it rains tomorrow, then I’ll stay home," or "Should I eat sukiyaki or moo krata?" Mastering this topic will help you think systematically and will definitely make grabbing those A-Level points much easier.
1. What are Propositions?
Before we start calculating, we need to know what a "proposition" is. A proposition is a declarative statement that can be clearly identified as either "True (T)" or "False (F)", but never both.
Examples of propositions:
- "1 + 1 = 3" (This is a proposition because we can identify it as False)
- "Bangkok is the capital city of Thailand" (This is a proposition because we can identify it as True)
Examples that are NOT propositions:
- "What should I eat today?" (A question)
- "Please do not make any noise" (A request/command)
- "He is so tall" (An exclamation, or a statement where it’s unclear who "he" refers to)
Key point: Anything that acts as a question, command, request, or a sentence containing a variable (e.g., x + 1 = 2 without defining what x is) is not a proposition.
2. Logical Connectives
In real life, we often combine sentences. In logic, we have four main connectives:
1) "And" Symbol: \( \land \)
Think of this as a condition where "both things must be true" to pass.
Memory Trick: Only "True and True" results in "True." Everything else is False.
2) "Or" Symbol: \( \lor \)
Think of this as having a choice where picking either one (or both) is fine.
Memory Trick: Only "False or False" results in "False." Everything else is True.
3) "If... then..." Symbol: \( \rightarrow \)
This is a conditional contract.
Memory Trick: There is only one case where the contract is broken: "True implies False" (T \( \rightarrow \) F) results in "False." Everything else is always True.
4) "If and only if" Symbol: \( \leftrightarrow \)
Think of this as a relationship where both sides must move in the same direction.
Memory Trick: "Same is True, Different is False."
- T \( \leftrightarrow \) T gives True
- F \( \leftrightarrow \) F gives True
- If one is True and the other is False, you get False.
5) "Negation" (Not) Symbol: \( \sim \)
This flips the truth value to the opposite. If it was originally True, adding a \( \sim \) makes it False immediately.
Truth Table Summary (The Heart of Logic!):
- \( T \land T \equiv T \) (Otherwise F)
- \( F \lor F \equiv F \) (Otherwise T)
- \( T \rightarrow F \equiv F \) (Otherwise T)
- \( Same \leftrightarrow Same \equiv T \) / \( Different \leftrightarrow Different \equiv F \)
3. Logical Equivalence
Equivalence means two propositions have the "same truth value in every case" and can be used interchangeably. The symbol used is \( \equiv \).
Common formulas for the A-Level exam:
1. Converting "If... then...": \( p \rightarrow q \equiv \sim p \lor q \) (Remember: Change the first part to a negation, change the sign to "or," and keep the second part the same.)
2. Contrapositive: \( p \rightarrow q \equiv \sim q \rightarrow \sim p \) (Remember: Swap the front and back, then negate both.)
3. De Morgan’s Laws: \( \sim(p \land q) \equiv \sim p \lor \sim q \) (Remember: Distribute the negation and change "and" to "or.")
4. Tautology
A tautology is a logical form that is "always true," regardless of whether the variables inside are true or false.
Easiest way to check for tautology:
For \( p \rightarrow q \), assume the whole statement is False (set the front to T and the back to F) and try to find a contradiction.
- If you find a contradiction \( \rightarrow \) it is a Tautology (it literally cannot be False).
- If you cannot find a contradiction \( \rightarrow \) it is not a Tautology.
Did you know?
The word "Tautology" implies a statement that is true by its own nature—it’s forever true!
5. Common Mistakes
- Confusion with \( p \rightarrow q \): Many people mistakenly think that if the front is False (F \( \rightarrow \)...), the result is False. Actually, if the front is False, the result is always True!
- Forgetting to distribute the negation: When you see \( \sim(p \lor q) \), don’t forget to change the sign in the middle from \( \lor \) to \( \land \).
- Misinterpreting Thai language: Words like "except" or "but" in logic are often equivalent to the "and" connector.
Key Takeaways
Basic Logic in A-Level Applied Mathematics 2 focuses on understanding truth values of connectives and determining equivalence:
1. Propositions must be clearly True or False.
2. Memorize this: And (must be both True), Or (must be both False to be False), If-Then (T to F is the only False case), If and only if (Same is True).
3. Equivalence: Front to back \( \equiv \) Not front or back.
4. Tautology: Always true in every case.
If you feel it's difficult at first, don't worry! Try practicing by drawing truth tables, and things will start to click. Logic is a very rewarding chapter for gaining easy marks. Keep it up, everyone!