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#deck:Math::If A then B
#tags:math logic converse
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In if A then B, what are A and B called?	A is the hypothesis (the assumption), B is the conclusion.	definition
When is if A then B false?	Only when A is true and B is false.	truth-table
If A is false, what is the truth value of if A then B?	True, whatever B is: the promise was never triggered, so it cannot be broken. Called vacuously true.	truth-table
What is the converse of if A then B?	If B then A.	definition
What is the inverse of if A then B?	If not A then not B.	definition
What is the contrapositive of if A then B?	If not B then not A.	definition
Which of converse, inverse, contrapositive always agrees with the original?	The contrapositive. It fails in exactly the same case: A true, B false.	truth-table
Which two of the four statements always agree with each other, apart from the original and contrapositive?	The converse and the inverse: the inverse is the contrapositive of the converse.	truth-table
If a statement is true, must its converse be true?	No. The converse fails in a different case and needs its own proof.	truth-table
What does A if and only if B mean?	If A then B, and if B then A: both the statement and its converse are true.	iff
How many proofs does an if and only if need?	Two, one for each direction.	iff
State the Pythagorean theorem with its converse as one if and only if.	A triangle is right if and only if a^2 + b^2 = c^2, with c its longest side.	iff
Why is every definition an if and only if?	It names exactly the things with the property, by agreement, so both directions hold without proof.	iff
Can examples prove an if A then B about infinitely many cases?	No. However many cases pass, the next could fail.	mistakes
How do you disprove if A then B?	Find one counterexample: a case where A is true and B is false.	mistakes
Known: if A then B. Observed: B. Conclude A?	No: affirming the consequent. It confuses the statement with its converse.	mistakes
Known: if A then B. Observed: not A. Conclude not B?	No: denying the antecedent. It confuses the statement with its inverse.	mistakes
Known: if A then B. Observed: not B. Conclude not A?	Yes: that is the contrapositive, which is valid.	mistakes
If it rained the street is wet. The street is wet. Did it rain?	Not necessarily: someone may have washed the street. Affirming the consequent.	mistakes
If n is divisible by 4 then n is even. Converse true?	No: 2, 6 and 10 are even and not divisible by 4.	problems
If n = 7 then n^2 = 49. Converse true?	No: n = -7 also has n^2 = 49.	problems
If x > 2 then x^2 > 4. Converse? Inverse?	Both false, with the same counterexample, x = -3.	problems
If a quadrilateral is a square, it has four right angles. Converse?	False: a 2 by 5 rectangle.	problems
If p is prime then 2^p - 1 is prime. True?	No: p = 11 gives 2047 = 23 x 89, after four cases that work.	problems
If 2^n - 1 is prime then n is prime. True?	Yes. Contrapositive: if n = ab, then 2^a - 1 divides 2^n - 1.	problems
Knowing only the Pythagorean theorem, a triangle has a^2 + b^2 not equal to c^2. Conclusion?	It is not right, by the contrapositive. No converse needed.	problems
Knowing only the Pythagorean theorem, a triangle has a^2 + b^2 = c^2. Can you conclude it is right?	Not from the theorem alone: that is affirming the consequent. The converse licenses it.	problems
Trap: the converse and the contrapositive are the same thing?	No. Converse: swap. Contrapositive: swap and negate. Only the contrapositive is equivalent to the original.	trap
Trap: if A then B is false whenever A is false?	No, it is true then. It is false only when A is true and B false.	trap
Trap: a true theorem has a true converse, since it is the same idea backwards?	No. Divisible by 4 implies even; even does not imply divisible by 4.	trap
Trap: 30 cases checked with no counterexample, so it is proved?	No. Only a general argument proves it; one counterexample would disprove it.	trap
Trap: the inverse of a true statement is true?	Not necessarily: the inverse agrees with the converse, not with the statement.	trap