(a) use the logical equivalences p → q ≡∼p ∨ q and p ↔ q ≡ (∼p ∨
q) ∧ (∼q ∨ p) to rewrite the given statement forms without using
the symbol → or ↔, and (b) use the logi- cal equivalence p ∨ q
≡∼(∼p∧ ∼q) to rewrite each statement
form using only ∧ and ∼.
* p∨∼q→r∨q
3. (10 pts.) Use logical equivalences to show that (p r)v(q r) and (pAq) r ane logically equivalent.
Problem 3.11 Show using a chain of logical equivalences that (p → r)A(q → r) pv q) →
Verify the logical equivalences using the theorem below: (p ∧ ( ~ ( ~ p ∨ q ) ) ) ∨ (p ∧ q) ≡ p Theorem 2.1.1 Let p, q, and r be statement variables, t a tautology, and c a contradiction. The following logical equivalences are true. 1. Commutativity: p1q=q1p; p V q = 9VP 2. Associativity: ( pq) Ar=p1qAr); (pVq) Vr=pv (Vr) 3. Distributivity: PA(Vr) = (p19) (par); p V (qar) = (pVg) (Vr) 4. Identity: pAt=p:...
In this assignment you will write code that will prove both equations for three logical equivalences (pick any three except the double negative law). Below is the list of logical equivalences. Please create a program that allows a user to test logical equivalences and have proof of their equivalency for the user. The rubric is below. Submit screen shots of the code, input, and output of the program. Theorem 2.1.1 Logical Equivalences Given any statement variables p, q, and r,...
. (25 points) Show each of these two statements are tautology or not, using Logical Equivalences and WITHOUT using Truth Table. If you use Truth Table, no marks will be assigned. 1. (p1-9) + (p+-9) 2. (p ) 9
(40 pts) Consider the transitivity of the biconditional: ((P HQ ) ^ ( Q R )) → ( P R ). a. Show that this argument is valid by deriving a tautology from it, a la section 2.1. You may use logical equivalences (p. 35), the definition of biconditional, and the transitivity of conditional: (( P Q) ^ (Q→ R)) → (P + R). Show that this argument is valid using a truth table. Please circle the critical lines. C....
Decide whether or not the following equivalences are valid. (a) [∀x,(P(x) ∨ Q(x))] ≡ [(∀x, P(x)) ∨ (∀x, Q(x))]. (b) [∃x,(P(x) ∨ Q(x))] ≡ [(∃x, P(x)) ∨ (∃x, Q(x))]. For each, demonstrate the equivalence or give a counterexample.
5 points Show that p + (q + r) and q + (pvr) are logically equivalent without using a truth table. To get full credit, include which logical equivalences you used.
Using propositional logic, write a statement that contains the propositions p, q, and r that is true when both p → q and q ↔ ¬r are true and is false otherwise. Your statement must be written as specified below. (a) Write the statement in disjunctive normal form. (b Write the statement using only the ∨ and ¬ connectives.
Assume that p NAND q is logically equivalent to ¬(p ∧ q). Then, (a) prove that {NAND} is functionally complete, i.e., any propositional formula is equivalent to one whose only connective is NAND. Now, (b) prove that any propositional formula is equivalent to one whose only connectives are XOR and AND, along with the constant TRUE. Prove these using a series of logical equivalences.