Negation

Week three came with another new set of information: negation.  Using the &not symbol, we can negate a statement, making the negation which is true when the statement is false.  To begin a negation, we start at the beginning of a statement, as is intuitive.  A statement beginning &notall x in X becomes &exist x in X, and vice versa.

The trickier part of negation comes after: the not symbol moves through the statement to the claims made about the variables initialized but the first part of the statement.  Negating a claim is different depending on what the claim is.  If it is an implication, the negation of that would be antecedent and not consequent. This makes sense if you think about what makes an implication false: the antecedent being true and the consequent being false.  Taking this:

P(x) => Q(x)

when negated becomes

P(x) &and &not Q(x)

When dealing with a conjunction (an and claim); the negation becomes an implication with a false consequent:

P(x) &and Q(x)

becomes

P(x) => &not Q(x)

Seeing this, it almost becomes that implication and conjunction are opposites, so long as you remember which elements must be false.  The final case would then be negating a disjunction (an or statement).  For this case, the disjunction becomes a conjunction between two false elements:

P(x) &or Q(x)

becomes

&not P(x) &and &not Q(x)

This gets more complicated when you get to implications with multiple variable interactions on either side.  I was not completely clear on how to deal with this until Week 4's Monday class, but I feel like I have a firm grasp on how to deal with these negations now.  When moving a negation through a convoluted statement, you have to move from the bigger claims down to the smaller ones the large ones are made up of.  For instance:

(P(x) &and Q(x)) => (R(x) &and S(x))

The first thing to negate is the central implication.  Doing this, we take the antecedent as true and negate the consequent:

(P(x) &and Q(x)) &and &not(R(x) &and S(x))

and then follow the negation through the consequent:

P(x) &and Q(x) &and (R(x) => &not S(x))

and voila! A negated statement with multiple claims.