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

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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.

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- 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 existential 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 prime*Bxy*mean(that is ). Prove that the following are*x*is a multiple of*y**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
*y*is free it maybe*true*or it maybe*false*depending on the state. It is clearly*true*if*y*=1, however it is*false*otherwise as for any*y*we can find an*x*which it doesn't divide, so the body is*false*in this case. - SOULTION: Since no occurence of
*x*is free, this is either*true*or*false*. This is*true*since the body is*true*when*x*is 2.

- SOLUTION: Since
- Give a bullet proof that (i.e. ).(9)
SOLUTION

Fri Apr 9 13:20:45 EDT 1999