Logic Lecture — Completely conquering ∧∨ combination conditionals — everything about complex a...
In earlier chapters on converse, inverse, and contrapositive, we learned the structure of the conditional statement A→B. We also practiced the basic truth tables for ∧ (AND), ∨ (OR), and ¬ (NOT).
Now in Ch3, we cover complex conditional statements that combine these two concepts. Forms where the antecedent (the first part) or consequent (the second part) contains ∧ or ∨ are the most frequently tested types in public enterprise and civil service logical reasoning exams. We will clearly organize the tricky parts step by step.
4 Core Patterns of Complex Conditional Statements
When ∧ or ∨ appears in the antecedent and consequent of a conditional statement, a total of 4 basic patterns are created.
| Pattern | Form | Description |
|---|---|---|
| Pattern 1 | (A ∧ B) → C | C is true when both A and B are true |
| Pattern 2 | (A ∨ B) → C | C is true if either A or B is true |
| Pattern 3 | A → (B ∧ C) | Both B and C are true if A is true |
| Pattern 4 | A → (B ∨ C) | At least one of B or C is true if A is true |
If you fully understand these four, you can solve most complex reasoning problems. Let’s analyze them one by one.
Build Your Own Truth Table
Swap the connectives (∧, ∨) and see how the truth values change.
Pattern 1: (A ∧ B) → C
“If A and B, then C”
This is a form with ∧ in the antecedent. C is guaranteed to be true only when A and B are simultaneously true.
Truth Table Analysis
| A | B | A∧B | C | (A∧B)→C |
|---|---|---|---|---|
| T | T | T | T | T |
| T | T | T | F | F (Condition violated) |
| T | F | F | T | T |
| T | F | F | F | T |
| F | T | F | T | T |
| F | T | F | F | T |
| F | F | F | T | T |
| F | F | F | F | T |
Core: If the antecedent (A∧B) is false, the conditional statement is unconditionally true. If either A or B is false, the antecedent is false, so the entire conditional statement is true.
Contrapositive Conversion: Contrapositive of (A ∧ B) → C
Contrapositive formula: ¬C → ¬(A ∧ B)
De Morgan’s laws: ¬(A ∧ B) = ¬A ∨ ¬B
Therefore, the contrapositive is: ¬C → (¬A ∨ ¬B)
That is, “If not C, then not A or not B.”
Practical Example
Given condition: “If the Korean score is 80 or higher (A) and the English score is 80 or higher (B), then you pass (C).”
- A∧B: Korean 80↑ and English 80↑
- (A∧B)→C: Both conditions met → Pass
Applying the contrapositive: If you fail (¬C) → Korean is below 80 (¬A) or English is below 80 (¬B).
Trap frequently tested on exams: “When (A∧B)→C is true, is A→C true?”
Answer: False. If only A is true and B is false, the antecedent is false so the conditional statement is true, but A does not guarantee C.
Pattern 2: (A ∨ B) → C
“If A or B, then C”
A form with ∨ in the antecedent. If either A or B is true, C must be true.
Core Point
This pattern is equivalent to the following two conditional statements:
- A → C
- B → C
This is because there are three cases where A∨B is true: (A only is true), (B only is true), and (both are true), and in any case, C must hold.
Therefore: (A ∨ B) → C ≡ (A → C) ∧ (B → C)
This is the most important equivalence transformation. Make sure to memorize it.
Contrapositive Conversion: Contrapositive of (A ∨ B) → C
¬C → ¬(A ∨ B)
De Morgan’s laws: ¬(A ∨ B) = ¬A ∧ ¬B
Therefore, the contrapositive is: ¬C → (¬A ∧ ¬B)
That is, “If not C, then neither A nor B.”
Practical Example
Condition: “If you overslept (A) or traffic was heavy (B), you are late (C).”
- (A∨B)→C: Even if just one of the two happens, you are late.
Separation: “If you overslept, you are late (A→C)” AND “If traffic was heavy, you are late (B→C)”
Contrapositive: If you weren’t late (¬C) → You didn’t oversleep (¬A) and traffic wasn’t heavy (¬B).
Pattern 3: A → (B ∧ C)
“If A, then B and C”
A form with ∧ in the consequent. If A is true, both B and C must be true.
Core Point
This pattern is equivalent to the following two conditional statements:
- A → B
- A → C
A → (B ∧ C) ≡ (A → B) ∧ (A → C)
Contrapositive Conversion: Contrapositive of A → (B ∧ C)
¬(B ∧ C) → ¬A
De Morgan’s laws: ¬(B ∧ C) = ¬B ∨ ¬C
Therefore, the contrapositive is: (¬B ∨ ¬C) → ¬A
That is, “If not B or not C, then not A.”
Practical Example
Condition: “When you submit a job application (A), you must submit both a cover letter (B) and a portfolio (C).”
Separation: “Submit job application → Submit cover letter” AND “Submit job application → Submit portfolio”
Contrapositive: If you did not submit a cover letter (¬B) or did not submit a portfolio (¬C) → The application is not accepted (¬A).
Pattern 4: A → (B ∨ C)
“If A, then B or C”
A form with ∨ in the consequent. If A is true, at least one of B or C must be true.
Contrapositive Conversion: Contrapositive of A → (B ∨ C)
¬(B ∨ C) → ¬A
De Morgan’s laws: ¬(B ∨ C) = ¬B ∧ ¬C
Therefore, the contrapositive is: (¬B ∧ ¬C) → ¬A
That is, “If neither B nor C, then not A.”
Practical Example
Condition: “If selected for the project (A), you will be assigned to Seoul or (B) Busan (C).”
- A → (B ∨ C): Selected → Seoul or Busan
Contrapositive: If neither Seoul (¬B) nor Busan (¬C) → You were not selected (¬A).
Caution: Just because B∨C is true does not mean A is necessarily true (Fallacy of the Converse).
4 Patterns Comprehensive Summary Table
| Original Conditional Statement | Equivalent Separation | Contrapositive |
|---|---|---|
| (A∧B)→C | - | ¬C→(¬A∨¬B) |
| (A∨B)→C | (A→C)∧(B→C) | ¬C→(¬A∧¬B) |
| A→(B∧C) | (A→B)∧(A→C) | (¬B∨¬C)→¬A |
| A→(B∨C) | - | (¬B∧¬C)→¬A |
Memorizing this table allows you to quickly convert any complex conditional statement.
De Morgan’s Laws: Two Formulas You Must Memorize
The core tools for all contrapositive conversions above:
- ¬(A ∧ B) = ¬A ∨ ¬B — Applying NOT to AND → changes to OR and negates each term
- ¬(A ∨ B) = ¬A ∧ ¬B — Applying NOT to OR → changes to AND and negates each term
Intuitive understanding:
- “Not (A and B)” = “Not A or not B” ✓
- “Not (A or B)” = “Not A and not B” ✓
Practical Problem Solving
Problem 1
When all of the following conditions are true, what is definitely true?
① (Selection for national project A ∨ Selection for national project B) → Budget increase ② National project A was selected.
Solution Process:
Applying Pattern 2: (A∨B)→C ≡ (A→C)∧(B→C)
Therefore, condition ① separates into: “A selected → Budget increase” AND “B selected → Budget increase”.
Since A is true according to condition ②, by Modus Ponens on “A → Budget increase”:
Conclusion: A budget increase occurs. ✓
Problem 2
When all of the following conditions are true, what is definitely true?
① If you participate in the project → you must submit both a proposal and a results report ② Manager Kim did not submit a results report.
Solution Process:
Applying Pattern 3: Contrapositive of A→(B∧C) = (¬B∨¬C)→¬A
“Did not submit results report (¬C)” → ¬(B∧C) → ¬A
Applying contrapositive: Manager Kim did not participate in the project. ✓
(Modus Tollens)
Problem 3 (Advanced)
When all of the following conditions are true, choose the one that must be true.
① If it rains (A) → cancel the picnic (B) or bring a raincoat (C) ② It rained (A). ③ The picnic was not canceled (¬B).
Solution Process:
① Pattern 4: A→(B∨C) From ② and ③: A=T, ¬B=T (i.e., B=F)
Since A is true, by ①, B∨C must be true. Since B is false, for B∨C to be true, C must definitely be true.
Conclusion: Brought a raincoat (C). ✓
This reasoning pattern is called Disjunctive Syllogism: (B∨C) and ¬B → C
Frequently Appearing Trap Types
Trap 1: Confusion between Affirming the Antecedent and Affirming the Consequent
In (A∧B)→C, if A is true, is C true?
False! If only A is true and B is false, the antecedent is false → the conditional statement is true, but we cannot say anything about C.
Trap 2: In (A∨B)→C, if C is true, is A∨B true?
False! Fallacy of affirming the consequent. C might be true for another reason. Even if the consequent is true, we cannot deduce that the antecedent is true.
Trap 3: Error of Converting Contrapositive to Converse
Converse of (A∧B)→C: C→(A∧B) — This is not equivalent.
Contrapositive: ¬C→(¬A∨¬B) — Only this is equivalent.
Core Summary: Just Remember This
- (A∨B)→C ≡ (A→C)∧(B→C): Antecedent OR → Split into each
- A→(B∧C) ≡ (A→B)∧(A→C): Consequent AND → Split into each
- De Morgan: ¬(A∧B)=¬A∨¬B / ¬(A∨B)=¬A∧¬B
- Contrapositive conversion involves swapping ∧↔∨
- Disjunctive Syllogism: (A∨B)∧¬A → B
In the next installment, Ch4: Chain Reasoning and Syllogism, we cover the principles of chain reasoning where A→B and B→C imply A→C, and how to solve long conditional chain problems using them.
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