Math 2090.03 N (Winter 98-99) TEST 1A
Friday February 12th, 1999
SURNAME (BLOCK CAPITALS):
All questions concern our propositional logic and material covered in Chapter 3 of the course text. The marks for each question are shown in
- Define sentence: NOTES (4)
- Let p,q,r be boolean variables. Are the following sentences? Give reasons. (6)
- SOLUTION No, missing a ).
- SOLUTION Yes p is a sentence, so is a sentence.
- SOLUTION No y is not a boolean variable.
- Fill in the to give a full dress proof that (). All
axioms, Metatheorems from Chapter 3, and lower numbered theorems may be used in this proof. (10)
Hence by TRANSITIVITY 3 times .
- Fill in the to give a full dress proof that (). All axioms, Metatheorems from Chapter 3, and lower numbered theorems may be used in this proof.
Hence by TRANSITIVITY 3 times, and EQUANIMITY we have that .
- Define soundness. NOTES Is sound? YES.(3)
- Define completeness. NOTES Is complete? YES. (3)
- Let p,q,r be boolean variables. Explain why is not a theorem in
SOLUTION In state (p,q,r)=(T,T,F) the sentence evaluates to F. However is
complete, and so the sentence cannot be a theorem.
- State Metatheorem 3.7. NOTES (2)
- Prove that if then . You may use anything from Chapter 3
except those facts concerning soundness and completeness. (8)
SOLUTION Assume . However , and substitution with p:= P gives
that . However , and so by Equanimity we have that . Metatheorem 3.7 states that any two theorems are equivalent, and in particular that . Leibniz with E being gives that , and so
by equanimity, .