Let P(x) and Q(x) be propositional functions. We wish to prove the following general equivalence:
Below is a derivation of this equivalence using step-by-step substitutions based on general equivalence laws that you have learned. Please fill in the missing steps.x (P(x)
Q(x))
(
x P(x))
| General rule used: | ||||
|---|---|---|---|---|
| 1. |
x (P(x) Q(x))
|
|
x (¬P(x) \/ Q(x))
|
[ pq ¬p \/ q ]
|
| 2. |
|
¬ x ¬(¬P(x) \/ Q(x))
| [ x: p ¬
x: ¬p ]
| |
| 3. | | __________________________ | [One of DeMorgan's laws.] | |
| 4. | | __________________________ | [ x: (p /\ q) ( x: p) /\ ( x: q) ]
| |
| 5. | |
¬ ( x P(x)) \/
¬ ( x ¬Q(x))
| __________________________ | |
| 6. | | __________________________ |
[ ¬
x: ¬q x: q ]
| |
| 7. | |
( x P(x))
(
x Q(x))
|
[ ¬p \/ q pq ]
|
Let A and B be sets of elements from a finite
universal set U. Re-write each of the following lists of
quantities in order of increasing size. (You can assume that A
and B and U are chosen such that all the values in each
list are different from each other.) A 
B is the set of elements that are
members of either A or B, but not both. Drawing Venn
diagrams may be helpful.
- a) |A|, |A
B|,
|A 
B|, |U|, |Ø|.
- ________, ________, ________, ________, ________
- b) |A - B|, |A B|,
|A| + |B|, |A B|, |Ø|.
- ________, ________, ________, ________, ________