Logical Implication

Context: FIT1058_MOC · the conditional connective $P\Rightarrow Q$ · equivalent to a subset containment · used by Modus Ponens

Quick Revision

• 🎯 Objective: $P\Rightarrow Q$: whenever $P$ holds, $Q$ holds ➔ forbids exactly one case — $P$ true, $Q$ false.
• 📦 Core Components: $P\Rightarrow Q\equiv \neg P\vee Q$ ➔ $\equiv P\subseteq Q$ ➔ biconditional = both directions.
• ⚡ Key Constraint: non-symmetric — the converse $Q\Rightarrow P$ does not follow.

📝 Core

1. The Conditional (One Forbidden Row)

• Meaning ➔ “if antecedent $P$ then consequent $Q$” ➔ forbids only $P$ T with $Q$ F.
• Domino ➔ (stands,stands), (stands,falls), (falls,falls) possible; only (falls,stands) impossible.
• Vacuous truth ➔ $P$ false ⟹ $P\Rightarrow Q$ true regardless of $Q$.

2. Two Rewrites

• Disjunction ➔ $P\Rightarrow Q\equiv \neg P\vee Q$ ➔ needs no special gate, just $\neg ,\vee$.
• Biconditional ➔ $P\iff Q\equiv (P\Rightarrow Q)\wedge (Q\Rightarrow P)$.

3. Implication as Containment

• Set view ➔ $P,Q$ as scenario-sets ➔ ”$P$ can’t happen without $Q$” ⟹ $P\setminus Q=\varnothing$.
• Equivalence ➔ $P\Rightarrow Q\equiv P\subseteq Q$ ➔ proving an implication = proving a containment.
• iff = set equality ➔ both directions ⟹ $P\subseteq Q$ and $Q\subseteq P$ ⟹ $P=Q$.

Key identities:

$P\Rightarrow Q\equiv \neg P\vee Q,P\Rightarrow Q\equiv P\subseteq Q$

⚖️ Core Decision Matrix

Statement	Formal	As sets
”if $P$ then $Q$” / ”$P$ only if $Q$"	$P\Rightarrow Q$	$P\subseteq Q$
"$P$ if $Q$” (converse)	$Q\Rightarrow P$	$Q\subseteq P$
”$P$ iff $Q$” (biconditional)	$P\iff Q$	$P=Q$
rewrite	$\neg P\vee Q$	complement ∪

When It Flips: $P\Rightarrow Q$ asserts neither $P$ nor $Q$ individually — only the conditional link. Implication ↔ subset is the bridge: proving containment proves implication and vice versa; double inclusion is iff.

📊 Exam Execution Trace

Manual Execution Trace

Verify $P\Rightarrow Q\equiv \neg P\vee Q$ and compare the converse:

Step / State	$P$	$Q$	$P\Rightarrow Q$	$\neg P\vee Q$	$Q\Rightarrow P$
0 (Init)	—	—	—	—	—
1	F	F	T	T	T
2	F	T	T	T	F
3	T	F	F	F	T
4	T	T	T	T	T

⚠️ Common Mistakes

• 💡 Beware the converse ➔ $P\Rightarrow Q$ gives no information about $Q\Rightarrow P$; seeing the right domino fallen doesn’t prove the left fell. ”$P$ only if $Q$” = $P\Rightarrow Q$; ”$P$ if $Q$” = $Q\Rightarrow P$.

🧠 Active Recall

Why is $P\Rightarrow Q$ equivalent to $P\subseteq Q$, and what is the one forbidden configuration?

• Hint: Scenario-sets.

Answer

• Short answer: No scenario has $P$ without $Q$ ⟹ $P\setminus Q=\varnothing$ ⟹ $P\subseteq Q$.
• Why: Forbidden row ➔ the only impossible case is $P$ true, $Q$ false (left domino falls, right stands).

What is the converse of $P\Rightarrow Q$, why can't you assume it, and what does the biconditional add?

• Hint: Non-symmetry.

Answer

• Short answer: Converse is $Q\Rightarrow P$; it doesn’t follow from $P\Rightarrow Q$.
• Why: Double inclusion ➔ both directions give $P\iff Q$ = $P=Q$ (iff / logical equivalence).