A-Level Mathematics Summary: Logic
Hello everyone! Welcome to the "Logic" chapter, one of the fundamental topics in Number and Algebra for the A-Level Applied Mathematics 1 exam. Think of this chapter as the "foundation of reasoning"—it’s essential not only for mathematics but also for your everyday life.
If you feel like logic sounds complicated with all these symbols, don’t worry! We’ll break it down into easy, bite-sized pieces, just like solving a jigsaw puzzle.
1. Propositions
First, we need to understand what a "proposition" is. A proposition is a declarative statement that is either "True (T)" or "False (F)", but never both.
Examples:
- "Chiang Mai is a province in Thailand." (Proposition - truth value is True)
- "\( 2 + 3 = 10 \)" (Proposition - truth value is False)
- "What would you like to eat today?" (Not a proposition because it’s a question; it cannot be labeled as true or false.)
Key point: Exclamations, commands, requests, or open sentences (statements containing variables that aren't defined) are not propositions.
2. Logical Connectives
We usually represent propositions with English letters like \( p, q, r \) and connect them using four primary logical connectives:
1) "And" - symbolized by \( \land \)
Think of this as a strict rule: It is only true if "both are true."
Easy trick: Only \( T \land T \) results in \( T \); everything else is \( F \)!
2) "Or" - symbolized by \( \lor \)
Think of this as being lenient: As long as there is at least one true value, it is true.
Easy trick: Only \( F \lor F \) results in \( F \); everything else is \( T \)!
3) "If...then..." (Implication) - symbolized by \( \rightarrow \)
Think of this as a "promise": If the premise is true but the outcome is false, the promise is broken!
Easy trick: \( T \rightarrow F \) is the only case that results in False (F); all other cases are true.
4) "If and only if" (Biconditional) - symbolized by \( \leftrightarrow \)
Think of this as "matching":
Easy trick: If they match, it’s true; if they differ, it’s false.
\( T \leftrightarrow T \) is true, \( F \leftrightarrow F \) is true
\( T \leftrightarrow F \) is false, \( F \leftrightarrow T \) is false
5) "Not" (Negation) - symbolized by \( \sim \)
This simply flips the truth value.
Easy trick: It’s always the opposite: \( T \) becomes \( F \), and \( F \) becomes \( T \).
Did you know? In logic, the symbol \( \lor \) (or) means "one or the other, or both." It’s different from when you order food and say, "Water or Orange Juice," which usually implies you can only choose one.
3. Tautology
A tautology is a compound proposition that is "True" in every possible case, regardless of the truth values of the individual propositions.
Common methods to check for tautology in exams:
1. Truth Table: If the last column is all \( T \), it’s a tautology (but this is time-consuming!).
2. "Prove it False" method (for \( \rightarrow \) or \( \lor \)): Assume the entire proposition is false, then work backward to find the values of \( p, q \). If you encounter a "contradiction," it means it is a tautology.
Common mistake: Students often confuse "contradiction" as meaning it's not a tautology. Remember: A contradiction = It is a tautology (because we failed to make it false).
4. Equivalence
Two propositions are equivalent (symbol \( \equiv \)) if they have the same truth value in every case.
Most frequent equivalence formulas (Must memorize!):
1. Convert "If...then": \( p \rightarrow q \equiv \sim p \lor q \)
2. Contrapositive: \( p \rightarrow q \equiv \sim q \rightarrow \sim p \)
3. De Morgan's Laws:
- \( \sim(p \land q) \equiv \sim p \lor \sim q \)
- \( \sim(p \lor q) \equiv \sim p \land \sim q \)
(Easy trick: Distribute the negation and flip the middle sign.)
5. Quantifiers
In this section, we encounter two unique symbols: \( \forall \) and \( \exists \).
1. \( \forall x \) (For All): Means "for every x."
- It is true: When every element in the universe of discourse (U) makes the statement true.
- It is false: As soon as there is "just one" that is false, the whole thing fails!
2. \( \exists x \) (For Some / There exists): Means "for at least one x."
- It is true: As long as there is "at least one" that makes the statement true, it works!
- It is false: It must be false for "every single one" to be considered false.
Negation of Quantifiers:
- \( \sim \forall x [P(x)] \equiv \exists x [\sim P(x)] \)
- \( \sim \exists x [P(x)] \equiv \forall x [\sim P(x)] \)
Easy trick: Swap \( \forall \) and \( \exists \), then place the \( \sim \) sign in front of the open sentence.
Key Takeaways
If it feels hard at first, don't worry. Try practicing by mastering these basic rules:
- \( \land \) (And) fears \( F \) (one \( F \) ruins it all).
- \( \lor \) (Or) loves \( T \) (one \( T \) saves it all).
- \( \rightarrow \) (If...then) has one weakness: \( T \rightarrow F \).
- \( p \rightarrow q \) can always be converted to \( \sim p \lor q \).
- Proving a tautology is essentially "trying your best to make it false"—if you can't, it is always true (it's a tautology).
Keep at it! Logic is a great way to secure marks. Once you understand the principles of connecting propositions, a great A-Level score is well within your reach!