Write a logical proof to prove that the following argument is valid. (t A S) → r (-g v p) → s .. r Vu

Write the negation of each of the following statements (hint: you may have to apply DeMorgan's Law multiple times) (a) ~ p^~ q (b) (p^q) → r

Consider the following finite-state automaton A over E = {a, b, c}: A = (E, {qo, 91; 92, 93 } , 8, 90, {92}) %3D where d = {(q0, a, q1), (q1, b, q0), (q1, b, q2), (q2, a, 91), (92, b, q2), (q2, a, q3), (q3, c, q3), (q3, a, q2)}. Draw the automaton A as a transition graph.

Given R=b*(ba+∪a+)+b*and S=(a*ba+b)∗ draw the automaton R and S

METHODS OF PROOF PROVE THE FOLLOWING: 1. By contradiction, prove that "for all real numbers x, if x is irrational, then -x is irrational. *Please provide a clear proof. THANK YOU.

Below are some statements; the domain of each is the set of all integers. For symbolic statements, thenegation symbol (¬) must not apear in your answer. For English statements, don’t use “It is not the casethat” or just the word “not” to form the negation.For each one, state whether the statement is TRUE, FALSE, or CAN’T BE DETERMINED. For all that onesthat are TRUE or FALSE, state the negation:1.∃n(nis even)2.∀a∀b(3a≥2b)3.∀a∃b(ab= 1)

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The correct statements are: A set is countable iff it is either finite or countably infinite. The set of natural numbers is countable. The power set of the set of natural numbers is countable. For any Σ Ø, Σ* is countably infinite.