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Question
I want to ask whether my understanding of mathematical statements and their converse, inverse, contrapositive is correct.
I'm trying to visualize these relations by Venn diagrams.

Am I  right in doing so?

Let's have statements P and Q.

So when I say P=>Q.
Does it mean: Whenever P is true, it implies Q is always true.

If it is, then I'm imagining P and Q to be 2 circles, like we've in Venn Diagrams, circle representing Q bigger and enclosing a smaller circle representing P.

We can't take P enclosing Q because then there will be some cases where P won't imply Q. So, Q encloses P and Q and P can't be congruent circles as they're different statements.

Now it is very easy to visualize some basic logical facts:

If statement is true then it's contrapositive is also true.
Explanation:
Since Q encloses P, contrapositive states that not Q => not P, which is obvious when we see from Venn diagram.

If converse is true then inverse of same statement is also true.
Explanation:
Converse : Q=>P is true then Inverse : not P=>not Q is actually contrapositive of Q=>P and this time considering that P encloses Q.


It is very easy to visualize through Venn diagrams...but is it the right way?

Thanks

Answer
Questioner:Apoorv
Country:India
Category:Advanced Math
Private:Yes  --- CORRECTED.
Subject: Symbolic logic.

Question:
I want to ask whether my understanding of mathematical statements and their converse, inverse, contrapositive is correct.
I'm trying to visualize these relations by Venn diagrams.

Am I  right in doing so?

>>  Looks OK to me.

Let's have statements P and Q.

So when I say P=>Q.
Does it mean: Whenever P is true, it implies Q is always true.

>> Actually it's a 'statement-form' with its own Truth Table, which is filled in like this:

[USE COURIER FONT TO VIEW THIS]
 p   q    p -> q
----+----+-------+
 T    T     T
----+----+-------+
 T    F   FALSE!!
----+----+-------+
 F    T     T
----+----+-------+
 F    F     T
----+----+-------+

>> and if you make these two statements:

1)  p -> q  is true
2)  p  is true

then you can conclude that:

3) q is true.

So yes, you are correct.  The above is important enough to be given not one, but TWO names:

Law of syllogism.
Modus ponens.

-----------------------------
If it is, then I'm imagining P and Q to be 2 circles, like we've in Venn Diagrams, circle representing Q bigger and enclosing a smaller circle representing P.

>>> I suggest for now, we use small letters for statements and caps for sets.

So small p shall mean:  x isin P
---------------------------------------------
'x isin P' is short for 'x is an element of P'
---------------------------------------------

Thus: p -> q means 'If x isin P, then x isin Q', which is really the definition of 'P is a subset of Q'.
------------------------------------------------------
We can't take P enclosing Q because then there will be some cases where P won't imply Q. So, Q encloses P and Q and P can't be congruent circles as they're different statements.

Now it is very easy to visualize some basic logical facts:

If statement is true then it's contrapositive is also true.

>>> you mean "its" with no apostrophe.

You can check this out by constructing the truth table for

-p  ->  -q
----------------------------------
Explanation:
Since Q encloses P, contrapositive states that not Q => not P, which is obvious when we see from Venn diagram.

If converse is true then inverse of same statement is also true.

>> because the converse and inverse are contrapositives of each other.

Explanation:
Converse : Q=>P is true then Inverse : not P=>not Q is actually contrapositive of Q=>P and this time considering that P encloses Q.

It is very easy to visualize through Venn diagrams...but is it the right way?

>> Looks OK to me.(Did I say that before?)
Thanks  

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