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The original proposition is (If I ace this exam, then i will be happy).Why is it option b rather option a when they both make sense.I know i can draw the truth table out to prove the logical equivalence but is there any way i can infer from the statements they are logically equivalent without a truth table.

a.(If i do not ace this exam, then i will not be happy) b.(If i am not happy, then i did not ace this exam)

NixyCron
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6 Answers6

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$$p\implies q$$ which we read as $ p $ implies $ q $, means that:

If $ p $ is true then certainly, $ q $ is true.

But if $ p $ is false, we can say nothing about $ q $.

If $ q $ is false, certainly $ p $ is false because if $ p $ was true, $ q $ will be true too.

$p$ and $ q$ are logically equivalent if they are both true or both false. In this case If $ p $ is false, then $ q$ will be false. we write $$p \iff q$$

hamam_Abdallah
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Suppose there are 2 things that can make you happy, acing the exam and finding a penny. Just because you fail doesn't mean you are unhappy, you might fail and still find a penny. But if you are unhappy, you neither aced the exam nor did you find a penny.

DanielV
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Think about it like this. Nowhere in the original statement is it implied that you won't be happy if you don't ace the exam. It only tells you what will happen if you do ace the exam.

So there's no way it could be option A, as we have no information from the original statement about what will happen if we don't ace the exam.

On the other hand, we know that if we aced the exam, we should be happy. Therefore with option B, because we know we're not happy, we can't have aced the exam.

Option B is in fact the contrapositive of the original statement (refer to the wikipedia article on contraposition).

Drathora
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a) is the inverse of the original statement, while b) is the contrapositive statement. One basic result of logic is that the contrapositive is ALWAYS equivalent to the original statement. You may read Velleman's How to Prove It.

toronto hrb
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  • By inverse you mean converse? – Junglemath Jun 26 '20 at 16:23
  • No. Consider "If P, then Q". The converse is "If Q, then P", while the inverse is "If not P, then not Q". – toronto hrb Jun 26 '20 at 16:28
  • Any homeschooler using Saxon Math should have learned this in its Algebra 1. Don't know whether public schools teach this or not. – toronto hrb Jun 26 '20 at 16:33
  • The contrapositive of (a) in OP is the converse of the original statement, and you claimed that (a) is the inverse. Hence by inverse you must mean converse. Also, "if Q then P" has contrapositive "if not P then not Q", which is what your inverse is in your example. So it sure seems like you're using inverse to mean converse. – Junglemath Jun 26 '20 at 16:35
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If you are not happy, then you didn't ace the exam, because if you did ace the exam, you'd be happy, which you aren't.

Junglemath
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Let $A$ and $B$ be logical propositions of unambiguous truth value at the moment. Then the logical implication $A\implies B$ being true means only that it is not the case that $A$ is true and $B$ is false.

$~~~~~A\implies B~~~\equiv~~~\neg [A \land \neg B]$

To answer your question, we let

$~~~~~A$ = My score on this exam is 100%

$~~~~~B$ = I am happy

$~~~~~A\implies B$

$~~~~~\neg [A \land \neg B]$

By the commutativity of $\land$, we have

$~~~~~\neg [\neg B \land A]~~~ \equiv \neg B \implies \neg A~~~~~$(corresponds to your option b)

Dan Christensen
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