Construct a truth table for each of the following compound propositions. a) p ⊕ (p ∧ q) b) (p ↔ q) ⊕ (p ↔ ¬q)

Use laws of logic to prove that (p ∧ q ∧ ¬r) ∨ (p ∧ ¬q ∧ ¬r) ≡ p ∧ ¬r.

1. Prove or refute: (∀x, y, z, w)(B(x, y, z) ∧B(w, y, z) →B(x, w, y)).2. Prove or refute: (∀x, y, z, w)(B(x, y, z) ∧B(x, w, y) →B(w, y, z)).

Consider the following predicates.  P(x) is the statement "x−4<1".Q(x) is the statement "x+4⩽3".R(x) is the statement "x−4>0". Determine the truth value of the following propositions.  P(8)→¬Q(−2) T or F? Q(3)∧¬R(3) T or F? R(5)→P(5) T or F?

Prove ⊢ (¬A → A) → A in Hilbert deductive system. Note: In addition to the axioms and rule of inference of H, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook). You may not use theorem 3.31, as this is precisely that theorem.book Mordechai Ben-Ari Mathematical Logic for Computer Science Third Edition Prove {¬A} ⊢ (¬B → A) → B in Hilbert deductive system. Note: In addition to the axioms and rule of inference of H, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook). book Mordechai Ben-Ari Mathematical Logic for Computer Science Third Edition PLEASE solve these with the help of 3 axioms and 1 rule of inference with the derived proof 3.20-3.30.

Construct proof for the following argument within the system of sentential logic: 1. (A & B) ⊃ (C V D)    Premise2. ~(C V (B ⊃ X))         Premise3. ~[D ≡ ~(X & Y)]        Premise4. ~A ⊃ ~Z                     Premise     /: .  ~Z

3. Use the Deduction Theorem to prove the following: ← (A → B) → (¬A → B) → B a. b. + ((A → B) → A) → A

Part 1: Choose two of the proofs below and use one of the indirect proof techniques (reductio ad absurdum or conditional proof) presented in Chapter 8 to demonstrate the validity of the arguments. The proofs below may use any of the rules of inference or replacement rules given in Chapter 8. 1.(G • P) → K, E → Z, ~P → ~ Z, G → (E v L), therefore, (G • ~L) → K 2.(S v T) ↔ ~E, S → (F • ~G), A → W, T → ~W, therefore, (~E • A) → ~G 3.(S v T) v (U v W), therefore, (U v T) v (S v W) 4.~Q → (L → F), Q → ~A, F → B, L, therefore, ~A v B 5.~S → (F → L), F → (L → P), therefore, ~S → (F → P) Part 2: Below are basic arguments in English. Choose two arguments and translate those argument into the symbolism of predicate logic. You do not need to do a proof. Every fetus has an immortal soul. A thing has an immortal soul only if it has a right to life. Hence, every fetus has a right to life. (Fx = x is a fetus, Sx = x has an immortal soul, Rx = x has a right to life). Some wars are just. No war of…

3. Express the following English sentences using propositional functions, quantifiers and logical connections. (a) Someone in the class is helped by everyone in the class. (b) There is a hard working student in the class who has not been helped by any student in class. (c) For everyone in the class, someone who is smart has helped him/her. SO (d) There are two people in the class who have not helped each other. (e) There is a hard working student in the class who is helped by every smart student in class. (f) There are some companies that hire only hard working students.

3. Prove ⊢ (¬A → A) → A in Hilbert deductive system. Note: In addition to the axioms and rule of inference of H, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook). You may not use theorem 3.31, as this is precisely that theorem.Theorem 3.20 ⊢ ¬A → (A → B). Theorem 3.21 ⊢ A → (¬A → B). Theorem 3.22 ⊢ ¬¬A → A. Theorem3.23 ⊢A→¬¬A.Theorem3.25 ⊢(A→B)→(¬B→¬A).Theorem3.28 ⊢(¬A→false)→A.Rule 3.29 (Reductio ad absurdum) Theorem3.30 ⊢(A→¬A)→¬A.   Prove {¬A} ⊢ (¬B → A) → B in H. Note: In addition to the axioms and rule of inference of Hilbert deductive system, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook).