Negation in the test!

Hi there dearest readers!

Hope everything is ok with you guys...

This week we had our midterm, I don't think I went bad in it, but there was a question in the test, if I'm not mistaken question 2, in which we had to negate a mathematical sentence, that for me was very confusing.

And it was not just negating, in that symbolic sentence we had to negate just the predicate.

Something like:

x  X , y  X , z  X, x X y = z

And in the exercise we had to negate just the predicate. So:

x X y = z

And my problem was, how can I negate a multiplication? In the test I just thought, ok, so it must be the division.

Researching for this post, I remembered a thing, so basic. That the "and" operator is a way of representing multiplication, so basically:

x X y = z      =      ((x  y)  z)

Translating the implication by its and/or form we arrive at:

(¬ (x  y)   z)

Negating this sentence by De Morgan's law we arrive at:

¬ ( (x  y)  ¬ z )

I think that is the negation of the first sentence, I just thought that would be interesting to present something in this form to its derivation. Please, if my calculations, or logic are wrong, comment! 

Negations, Converses and Contrapositives

Hi there, dear reader! Hope everything is all right!

When I was in my University back in Brazil, I was never introduced to the concepts of Converses and Contrapositives in formal logic systems. I think the concept of changing a thing on just on perspective quite nice, much more nice, when such thing its not a literal negation.

Processing Negation:

Let's suppose the following statement is true:

x  X | P(x)  Q(x)

Then the Negation of that statement is:

 (x  X | P(x)  Q(x))

The quantifier is changed and the negation go inwards:

 x  X | (P(x)  Q(x))

Then, in order to complete the process of negation we have to convert the implication to its and/or form which becomes:

 x  X | (P(x)  Q(x))

Finally we arrive at the negation of the following sentence:

x  X | P(x)  Q(x))

That is, of course the negation of the first sentence!

One easy way to discover if a sentence is a Converse, Contrapositive or the Negation of some sentence is to look for the quantifier. If the quantifier is UNCHANGED then it is the Converse or the Contrapositive, since its properties, just change the relationship between implications, ands or ors.

The Contrapositive of the first sentence would be:

x  X | Q(x)  P(x)

And the Converse of the first sentence would be:

x  X | Q(x)  P(x)

In some sense, you can say that these sentences derived from the original represent some kind contradiction in relation to the first one but NOT its negative perspective. I just thought this would be a nice addition to the blog.

Hope everyone like it!