Exercises identified by number are from the Gries & Schneider text. Of course, in proving a numbered theorem, you may use only lower-numbered theorems.
A1. Do Exercise 9.2, which can be restated as follows:
Take (∀x |: P) ∧ (∀x |: Q) ≡ (∀x |: P∧Q) to be an axiom. Without using Theorem (8.15), and using no theorem numbered higher than (9.2), prove
A2. Do Exercise 9.6, which can be restated as follows:
Prove (∀x | R∨Q : P) ≡ (∀x | R : P) ∧ (∀x | Q : P)
without using Axiom (8.18). (You probably will want to use Axiom (8.15), however.)
A3. Do Exercise 9.15, which is to prove
Of course, you will need to use Axiom (9.17) and/or Theorem (9.18).
A4. Do Exercise 9.23, which is to prove (∃-quantification) Range weakening/strengthening (9.25):
Of course, you will need to use Axiom (9.17) and/or Theorem (9.18). Other theorems likely to be helpful are (∀-quantification) Range weakening/strengthening (9.10) and (3.61) (Contrapositive).
Translate each of the following (English) statements into the language of predicate logic:
B1. There is a male who is liked by no one but himself.
B2. Every male likes at least two different females.
B3. Every person likes someone who does not like her/him back.
B4. There is no person who is liked by everybody.
B5. There are two females who like the same male but do not like
each other.
B6. If there is a person who is liked by everyone, then there is
also a person who is liked by no one except her/himself.
B7. Every female likes some male who is disliked by all her
female friends. (Note: The statement
"x and y are friends" translates to L.x.y ∧ L.y.x.)