Math 2090.03 N (Winter 98-99) TEST 1B
Friday February 12th, 1999
SOLUTIONS

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All questions concern our propositional logic and material covered in Chapter 3 of the course text. The marks for each question are shown in brackets.

1. Define theorem (in ): NOTES (6)
2. Let p,q,r be boolean variables. Give reasons why the following are theorems in . (4)
1. SOLUTION Axiom 3.3.
2. SOLUTION Axiom 3.28, Subs p:=q.
3. SOLUTION Axiom 3.57.
1. 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, and EQUANIMITY, we have that .

2. 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 .

1. Define completeness. NOTES is not complete. (Yes/No) NO.(3)
2. Define soundness. NOTES is not sound. (Yes/No) NO.(3)
3. Let p,q,r be boolean variables. Explain why is not a theorem in .(4)

SOLUTION In state (p,q,r)=(T,F,F) the sentence evaluates to F. However is complete, and so the sentence cannot be a theorem.

1. State Metatheorem ``Redundant True''. NOTES (2)
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 by Metatheorem ``Redundant True'' and 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, .