Statement : If it is raining, then the home team wins.
Converse : If the home team wins, then it is raining.
Why are these two not logically equivalent? The statement says that if it rains then the home team wins. In converse, since the home team wins, it should rain. Right?
It may help to ask yourself what would have to happen for each of the statements to be false.
(A) If it is raining, then the home team wins.
The only way this can be false is for it to be raining, and the home team loses. (For the same of simplicity I’ll assume that there are no ties.)
(B) If the home team wins, then it is raining.
The only way this can be false is for the home team to win while it is not raining.
Those are two completely different events:
- It’s raining, and the home team loses.
- It’s not raining, and the team wins.
If the two statements were equivalent, exactly the same circumstances would render them both false. As you can see, that’s not the case, so they cannot be equivalent.
In fact, there are four possible states of affairs:
- It’s raining, and the home team loses.
- It’s not raining, and the team wins.
- It’s raining, and the home team wins.
- It’s not raining, and the home team loses.
As we’ve seen, statement (A) is inconsistent with state (1); what about the other three states? It’s obviously consistent with state (3). It’s true but less obvious that it’s also consistent with states (2) and (4): this is because statement (A) says nothing at all about who wins if it’s not raining, so no outcome of the game can disprove it in that case. Thus, statement (A) boils down simply to saying that state (1) isn’t the case: either it’s not raining, in which case (A) says nothing about the outcome of the game, or it is raining, and the home team wins.
I’ll leave it to you to apply similar reasoning to see that statement (B) is consistent with each of the states except state (2).
The original statement says that if it’s raining, the home team wins. It is also possible, based on this statement, that even if it’s not raining, the home team still wins.
The converse statement says that if the home team wins, then it’s raining. It is also possible for it to rain without the home team winning.
However: Under this converse, it is not possible for the home team to win without it raining. Conversely (!), under the original statement, it is not possible for it to be raining without the home team winning.
Alternatively, you can show logical equivalence with a truth table. Let $r = textRains, w = textHome team wins$. Then
beginarrayc r&w&rto w&wto rhline T&T&T&Thline T&F&F&Thline F&T&T&Fhline F&F&T&T endarray Notice that the conditional statements $rto w$ and $wto r$ have different truth tables, and hence, they are not logically equivalent.
Why is statement and its converse not equivalent? – math.stackexchange.com #JHedzWorlD
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