3.4 Prove the following metatheorem. Q = true is a theorem iff Q is a theorem....

Transcribed Image Text:

3.4 Prove the following metatheorem. Q = true is a theorem iff Q is a theorem. 3.6 Assume that operatoris identified with operator of Sec. 2.1 (see Ex- ercise 3.1) and true is identified with the symbol true of Sec. 2.1. Prove that axioms (3.8) and (3.9) uniquely define operator. That is, determine which of the four prefix operators o defined in the truth table on page 26 satisf false otrue and o(p= q) = op=q. (3.8) Axiom, Definition of false false = true (3.9) Axiom, Distributivity of over: - (p=q) = p=q t t t f ← III n # FREC → t t t f f f t t a n = A d t f ft V tt t t t t t t t t f f t f Note: true is abbreviated by t and false by f # f f f t t HOT n r f f f f f f f t f f f 3.1 We have defined using three axioms. Assuming that the symbol true is identified with the symbol true of the previous chapter on boolean expressions, do the axioms uniquely identify operator = ? Answer this question by seeing which of the 16 possible binary operators o (say) given in the truth table on page 26 satisfy ((poq)or) o (po (qor)), pogoqop, and true o qoq. (For example, the operator given by the last column does not satisfy true ogoq, since the operator always yields f.) 3.4 Prove the following metatheorem. Q = true is a theorem iff Q is a theorem. 3.6 Assume that operatoris identified with operator of Sec. 2.1 (see Ex- ercise 3.1) and true is identified with the symbol true of Sec. 2.1. Prove that axioms (3.8) and (3.9) uniquely define operator. That is, determine which of the four prefix operators o defined in the truth table on page 26 satisf false otrue and o(p= q) = op=q. (3.8) Axiom, Definition of false false = true (3.9) Axiom, Distributivity of over: - (p=q) = p=q t t t f ← III n # FREC → t t t f f f t t a n = A d t f ft V tt t t t t t t t t f f t f Note: true is abbreviated by t and false by f # f f f t t HOT n r f f f f f f f t f f f 3.1 We have defined using three axioms. Assuming that the symbol true is identified with the symbol true of the previous chapter on boolean expressions, do the axioms uniquely identify operator = ? Answer this question by seeing which of the 16 possible binary operators o (say) given in the truth table on page 26 satisfy ((poq)or) o (po (qor)), pogoqop, and true o qoq. (For example, the operator given by the last column does not satisfy true ogoq, since the operator always yields f.)

Answer rating: 100% (QA)

To prove that axioms 38 and 39 uniquely define the operat... View the full answer