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# why inverse of a conditional statement seems to be TRUE?

Posted on 2012-09-21
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Consider following conditional statment:

If two points are distinct, then there is exactly one line that passes through them.

Inverse is as follows:

If two points are not distinct, then it is not true that there is exactly one line that passes through them.

According to high school geometry book, inverse is always FALSE but in above example inverse seems to be TRUE.  If both points are same, then, more than one line can pass through them.
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Question by:naseeam

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Expert Comment

correction,
if two points are distinct, then they are the same points

point a = 3,3
point b = 3,3

there can be many lines that pass through

so any line going through 3,3 is true.

the inverse would be a mirror line   on the opposite quardant, thus it will not pass through
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LVL 84

Accepted Solution

> According to high school geometry book, inverse is always FALSE
It seems that either  high school geometry book is incorrect, or you have misinterpreted what it said.
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Assisted Solution

http://www.mathwords.com/i/inverse_conditional.htm

If a statement is true, its inverse may or may not be true.

==========================================
If he dies today, he will be dead tomorrow.   TRUE

If he doesn't die today, he won't be dead tomorrow.    But what if he died yesterday?
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Author Comment

> It seems that either  high school geometry book is incorrect, or you have misinterpreted what it said.
Here is the link to the book:

http://www.nexuslearning.net/books/ML-Geometry/Chapter2/ML%20Geometry%20Chapter%202%20Review-Test.pdf

Please look at equivalent statements on page 72.
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Assisted Solution

The inverse always has the same truth value as the converse.

If inverse is NOT true JUST BECAUSE the conditional is true.

http://www.mathplanet.com/education/geometry/proof/if-then-statement

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Author Comment

> if two points are distinct, then they are the same points
I disagree.
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LVL 15

Expert Comment

pardon, meant "not-distinct"

typo
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LVL 13

Assisted Solution

You have mis-interpreted the book.

The book is not saying that the inverse is always opposite the conditional.

it says the inverse is equal to the converse and that the contrapositive is equal to the conditional.
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LVL 15

Expert Comment

>>If two points are not distinct, then it is not true that there is exactly one line that passes through them.

if two points ARE distinct, then it is True that there is exactly one line that passes through them.
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LVL 13

Assisted Solution

The Truth table for this statement is

conditional: TRUE
converse: TRUE  if exactly one line passes through two points, then they are distinct
contrapositive: TRUE if not exactly 1 line passes through two points, then they are not distinct
inverse: TRUE.if two points are not distinct, then its not true that exactly one line passes through them
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LVL 84

Expert Comment

While I see the book giving an example of an inverse that is false, I see nowhere any claim that inverse is always false.
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Author Closing Comment

Superb job!

Incredible excellent answers in such a short time.

Genuis minds!
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