Logical Reasoning — Completely conquering necessary conditions, sufficient conditions, inverse...
In logical reasoning, the converse, inverse, and contrapositive of a proposition are core concepts in both logic and mathematics exams. In particular, the contrapositive is always equivalent to the original proposition (it has the same truth value), making indirect proof by contrapositive a powerful tool when direct proof is difficult.
Basic Proposition: p → q
The most basic proposition form in logic is “if p, then q (p → q).”
- p: antecedent, assumption
- q: consequent, conclusion
Example: “If it rains (p), the ground gets wet (q).”
Converse, Inverse, and Contrapositive
From the original proposition p → q, we can derive three related propositions.
| Name | Symbol | Form | Relationship to the original proposition |
|---|---|---|---|
| Converse | q → p | “If q, then p” | Not equivalent |
| Inverse | ¬p → ¬q | “If not p, then not q” | Not equivalent |
| Contrapositive | ¬q → ¬p | “If not q, then not p” | Equivalent |
Key Relationships
- p → q ≡ ¬q → ¬p (original proposition ≡ contrapositive)
- q → p ≡ ¬p → ¬q (converse ≡ inverse)
Important: An original proposition and its converse are not necessarily equivalent.
Understanding Through an Example
Original proposition: “If it is a dog (p), it is an animal (q).” (p → q, true)
| Proposition | Form | Truth value |
|---|---|---|
| Original proposition | If it is a dog, it is an animal | True |
| Converse | If it is an animal, it is a dog | False (cats are animals too) |
| Inverse | If it is not a dog, it is not an animal | False (a cat is not a dog but is an animal) |
| Contrapositive | If it is not an animal, it is not a dog | True (equivalent to the original proposition) |
→ When the original proposition is true, its converse and inverse may not be true. → If the original proposition is true, its contrapositive must also be true.
Necessary and Sufficient Conditions
In the proposition p → q:
| Role | Definition | Example |
|---|---|---|
| p is a sufficient condition for q | If p holds, q is guaranteed | If it is a “dog,” it must be an “animal” |
| q is a necessary condition for p | If q does not hold, p cannot hold either | If it is not an “animal,” it cannot be a “dog” |
Necessary and Sufficient Condition (Biconditional, ↔)
p ↔ q (p is a necessary and sufficient condition for q, p if and only if q):
- p → q and, at the same time, q → p
- In other words, if p then q, and if q then p
Example: “If x = 2, then x² = 4” (true) However, “If x² = 4, then x = 2” (false — x = -2 is also possible) Therefore, “x = 2” is a sufficient condition for “x² = 4,” but not a necessary and sufficient condition.
Mnemonic for Necessary and Sufficient Conditions
When p → q
- p is a sufficient condition for q — p sufficiently guarantees q
- q is a necessary condition for p — without q, p cannot exist, so q is necessary
Common Question Types
Question 1: Using the Contrapositive
If “All philosophers study logic” is true, which of the following must be true?
① Everyone who studies logic is a philosopher ② If someone does not study logic, that person is not a philosopher ③ If someone is not a philosopher, that person does not study logic ④ There is a philosopher who studies logic
Solution: Original proposition: “Philosopher (p) → studies logic (q)” [true]
- ①: q → p (converse) → not equivalent, not necessarily true
- ②: ¬q → ¬p (contrapositive) → equivalent to the original proposition, necessarily true ★
- ③: ¬p → ¬q (inverse) → not equivalent, not necessarily true
- ④: Derived from the original proposition but has different content
Answer: ②
Question 2: Identifying Necessary and Sufficient Conditions
Which of the following is correct?
① “x = 3” is a necessary and sufficient condition for “x² = 9” ② “x > 0” is a sufficient condition for “x² > 0” ③ “being a triangle” is a necessary condition for “being a polygon”
Solution:
- ①: x = 3 → x² = 9 (true), x² = 9 → x = 3 (false, because x = -3 is also possible) → not a necessary and sufficient condition
- ②: x > 0 → x² > 0 (true) → x > 0 is a sufficient condition for x² > 0 ✓ But: x² > 0 → x > 0? False (x = -1 also gives x² = 1 > 0) → it is not a necessary condition
- ③: “being a triangle” → “being a polygon” (true) → a triangle is a sufficient condition for a polygon “being a polygon” is a necessary condition for “being a triangle” ✓
Answer: ② or ③ (multiple answers may be possible depending on the question context)
Indirect Proof Using the Contrapositive
When it is difficult to prove p → q directly, prove the contrapositive ¬q → ¬p.
Example: Prove that “if x² is odd, then x is odd.”
Direct proof: Difficult (it is hard to find every x for which x² is odd)
Proof by contrapositive: “If x is even, then x² is even.”
Assume x = 2k. Then x² = (2k)² = 4k² = 2(2k²) → x² is even (proof complete!)
If the contrapositive ¬q → ¬p is true, the original proposition p → q is also true ✓
Converse, Inverse, and Contrapositive in Compound Propositions
When a proposition includes multiple conditions:
Original proposition: “If A and B, then C” [(A ∧ B) → C]
- Converse: If C, then A and B [C → (A ∧ B)]
- Inverse: If not A or not B, then not C [¬(A ∧ B) → ¬C]
- Contrapositive: If not C, then not A or not B [¬C → ¬(A ∧ B)]
Watch De Morgan’s law: ¬(A ∧ B) = ¬A ∨ ¬B
Converse, inverse, and contrapositive are central to propositional logic. The fact that the contrapositive is always equivalent to the original proposition is used broadly, from mathematical proofs to LSAT logic questions and college entrance mathematics. In the next article (Ch6), we will cover quantifier logic (all, some) and how to negate universal and existential propositions.
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