Advanced Topics in Mathematics – Logic and Metamathematics
Mr. Weisswange
Assignment #6
Logic
1. Analyze the logical forms:
(a) Nobody in the calculus class is smarter than everybody in the discrete math class.
(b) Everyone likes Mary, except Mary herself.
(c) Jane saw a bear and Roger saw one too.
(d) Jane saw a bear and Roger saw it too.
(e) If anyone can do it, Orville can.
(f) If Orville can do it, anyone can.
2. Analyze the logical forms of the following statements. The universe of discourse is . What are the free
variables in each statement?
(a) Every number that is larger than x is larger than y.
(b) For every number a, the equation ax 2  4 x  2  0 has at least one solution iff a  2 .
(c) All solutions of the inequality x3  3x  3 are smaller than 10.
3. Translate the following statements into idiomatic English:
(a) x  H  x    y ~ M  x, y    U  x   , where H(x) means x is a man, M(x, y) means x is married to y, and
U(x) means x is unhappy.
(b) z  P  z, x   S  z, y   W  y   , where P(z, x) means z is a parent of x, S(z, y) means z and y are siblings,
and W(y) means y is a woman.
4. Are these statements true or false? The universe of discourse is .
(a) x y  2 x  y  0
(b) y x  2 x  y  0
(c) x y  x  2 y  0 
(e) y z  y  z  100
(d)  x  x  10   y  y  x  y  9  
(f) x y  y  x  z  y  z  100 
Metamathematics
Read Chapter V, GEB pp. 127-152.
5. Find an example of recursion in popular culture (in a movie, a TV show, popular novel, etc.) Bonus points if
you can bring in a YouTube clip name (that I can pull up in class) showing the recursion.
6. Try to follow the following list of sentences. How far down do they continue to make sense to you? If you
said “d”, do you program computers regularly?
(a) The mouse ate the cheese.
(b) The mouse the cat chased ate the cheese.
(c) The mouse the cat the dog bit chased ate the cheese.
(d) The mouse the cat the dog the woman yelled at bit chased ate the cheese.
7. The answer to this question is 7 less than 5 times the answer to question 7.