**Math 2090.03 N (Winter 98-99) TEST 2B Part 2
Friday April 9th, 1999 SOLUTIONS**

SURNAME (BLOCK CAPITALS):

INITIALS:

STUDENT NUMBER:

SIGNATURE:

\

All questions concern our predicate logic and material covered in Chapters 8 and 9 of the course text. The marks for each question are shown in brackets. You may NOT use inference rule/Metatheorem MON-AMON or inference rule/Metatheorem MODUS-PONENS anywhere on the test.

All axioms, allowed Metatheorems from Chapters 3, 4, 8, 9, and lower numbered theorems may be used in the proof of a higher numbered theorem.

The 4th page has been left blank for working, please indicate if there is part of your test there that needs marking. You are advised to do the questions worth the most marks first.

\

- Translate the following english sentences into quantified expressions in predicate logic. First
construct suitable types, then a suitable 2-place predicate, and then write the quantified expressions.
However, in this question you may ONLY use the universal quantifier, , in your translation.
(3)
- Not all cars have four wheels.
SOLUTION: Let
*C*be type cars, and*Fx*mean*x*has 4 wheels.

- No mouse is heavier than any elephant.
SOLUTION: Let
*M*be type mouse,*E*be type elephant,*Hxy*mean.*x*is heavier than*y*

- Not all cars have four wheels.
SOLUTION: Let
- Compute the following contextual substitution: (2)

SOLUTION:

- Let our UD be . Let
*Ax*mean, and*x*is positive*Bx*meanand*x*is prime*Cx*mean. Prove that the following are*x*is odd*true*,*false*, or may be*true*in a certain state, and may be*false*in another. In the latter case**if possible**give a state for which it is*true*and a state for which it is*false*, otherwise justify why all states either give*true*or give*false*. (6)- SOLUTION: Since all occurrences are bound this is
either
*true*or*false*. It is*false*since if*x*is 2, then the body is*false*. - SOLUTION: Since all occurrences are bound
this is either
*true*or*false*. It is*true*since if*x*is 4 then the body is*true*.

- SOLUTION: Since all occurrences are bound this is
either
- Give a bullet proof that (i.e. ).(9)
SOLUTION

Fri Apr 9 13:18:58 EDT 1999