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- Prove the following logical implication (A ⇒ B) ⇒ ((C ⇒ A) ⇒ (C ⇒ B)) without a truthtable.Draw a truth table to find the truth value of the following law of equivalence.Recall that PA has as axioms P 1, · · · P 6 and the induction scheme. Show that P A ⊢ ∀x∀y(x · (sy) = x · y + x) First give the explicit instance of the induction scheme that you need. Then show how that induction axiom is used to establish the result. You do not need to give an LK proof, but justify each step of your argument with one of the axioms (P1, ..., P6 or whatever is appropriate). You may take equality reasoning for granted.
- Reduce the proposition using laws Simplify q∨(w∧¬w) to q Select a law from below to apply to: q∨(w∧¬w) Laws Distributive (a∧b)∨(a∧c)≡a∧(b∨c) (a∨b)∧(a∨c)≡a∨(b∧c) Commutative a∨b≡b∨a a∧b≡b∧a Complement a∨¬a≡T a∧¬a≡F Identity a∧T≡a a∨F≡a Double negation ¬¬a≡aAnswer the following questions with adequete proofs:Simplify the following expression (Boolean Algebra) such that WXY will be the only remaining terms and Z will be cancelled in the expression. A = W'XYZ + WX'YZ + WXY'Z + WXYZ' + WXYZ
- Show that the argument is valid, indicating the rule of inference or logical equivalence applicable to each step. ∃x ( C(x) ∧ ( ¬ S(x) → ( V(x) ∨ W(x) ) ) )∀x ( V(x) → ¬ C(x) )¬ ∃x ( W(x) ∧ C(x) ) _________________________∴ ∃x ( S(x) ∧ ¬ W(x) )Consider the effect that the proliferation of personal computers and the Internet have had on the design of computer systems to put this into perspective. Make sure that your claims are supported with proof, if that is at all feasible.Reduce the proposition using laws. Simplify (q∨p)∧(¬p∨q) to q Select a law from below to apply to: (q∨p)∧(¬p∨q) Laws Distributive (a∧b)∨(a∧c)≡a∧(b∨c) (a∨b)∧(a∨c)≡a∨(b∧c) Commutative a∨b≡b∨a a∧b≡b∧a Complement a∨¬a≡T a∧¬a≡F Identity a∧T≡a a∨F≡a Double negation ¬¬a≡a
- Prove by Systematic Semantic Tableaux ∃xR(x,x)∨∃x[R(x,x)→¬∃yR(y,x)]Using logical Equivalence Laws, show that -a) ¬(¬p ∧ q) ∧ (p ∨ q) is logically equivalent to p.b) ¬(p ∨ ¬( p ∧ q )) is logically equivalent to F (i.e. a contradiction).Simplify the following Boolean expressions to a minimum number of literals indicating theorems used for each step . 1. ABC + A 'B + ABC' 2. x'yz + xz 3. (x + y)'(x' + y') 4. xy + x(wz + wz') 5. (BC' + A' D)(AB' + CD ')