This week, we tackled many means of representing the same logical statement with a different combinations of symbols. One rule that I found especially interesting was DeMorgan's Law; it simply shows the effect of negation on logical symbols, including a strange effect on both conjunction and disjunction. I found that making visual representations helps.
For the following example, we shall assume that A(x) represents the set of people who like apple pies, and C(x) will represent the set of people who prefer cherry pies.
I'm quite sure there are many of us who would devour both with equal enthusiasm, and as such, this example will start out with a simple statement to represent the likes of us;
A(x) ∧ C(x). (Figure 1).
A(x) ∧ C(x). (Figure 1).
(For the sake of simplicity, I will omit quantifiers in this representation.)
Negating the above:
¬ (A(x) ∧ C(x)). (Figure 2)
¬ (A(x) ∧ C(x)). (Figure 2)
As we already know, the negation sign "bubbles" through the statement until it hits the closing bracket, messing with virtually everything in its way. After all constituents have been negated, we get something familiar, but with a twist:
¬ A(x) ∨ ¬ C(x). (Figure 3).
¬ A(x) ∨ ¬ C(x). (Figure 3).
The intersection, or is know as the conjunction, becomes a disjunction, uniting the negated sets! It is also important to note that negating a disjunction yields a conjunction much in the same way.
In conclusion, DeMorgan's Law shows how the statement with the negation lingering outside of the brackets is equivalent to the statement with the negation "bubbled" in to affect every element, including conjunction and disjunction symbols.
