Implication, contrapositive, converse
Implication and its variants: table, necessary and sufficient conditions, converse, contrapositive, negation, and the two classic invalid arguments.
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Implication p ⇒ q: what is it?
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A proposition that is false in a single case: p true and q false; p is called the antecedent and q the consequent.
When the antecedent is false, the implication is true whatever q is: "if 2 = 3, then the Moon is made of cheese" is true, because the promise is never put to the test.
For a real number x, what is the contrapositive of "if x > 2, then x² > 4"?
- If x² ≤ 4, then x ≤ 2
- If x² > 4, then x > 2
- If x ≤ 2, then x² ≤ 4
- If x² < 4, then x < 2
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If x² ≤ 4, then x ≤ 2
Swap and negate: the opposite of x² > 4 is x² ≤ 4, that of x > 2 is x ≤ 2. "If x² > 4, then x > 2" is the converse, false for x = −3; "if x ≤ 2, then x² ≤ 4" is the inverse; the strict inequalities leave out the equality cases.
An implication whose antecedent is … is always true, whatever its consequent.
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An implication whose antecedent is false is always true, whatever its consequent.
p ⇒ q ≡ ¬p ∨ q: if p is false, ¬p is true and the ∨ is true.
Converse: what is it?
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The implication obtained by swapping antecedent and consequent: for p ⇒ q, it is q ⇒ p.
There is no reason for it to be true at the same time: "if n is a multiple of 4, then n is even" is true, but "if n is even, then n is a multiple of 4" is false (n = 2).
A regulation reads: "To obtain the degree, the internship must be passed." What does passing the internship guarantee?
- Nothing about the degree: it is a necessary condition, not a sufficient one
- The degree: it is a sufficient condition
- The degree, but only with the other modules
- The degree: in a regulation, "must" means "is enough"
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Nothing about the degree: it is a necessary condition, not a sufficient one
The rule reads "degree ⇒ internship passed": the internship is necessary. Without it, no degree, but having it is not enough, other requirements may remain.