Given the following premises: (1) ∀x (A(x) → B(x)) (2) ∀x (B(x) → C(x)) Prove the conclusion ∀x (A(x) → C(x)) using the rules of first-order logic. Show your step-by-step reasoning.

Given the following premises: (1) ∀x (R(x) → S(x))(2) ¬S(c) Use modus tollens to derive the conclusion ¬R(c). Show your step-by-step reasoning.

Prove the language L = {1n | m ∈ N ∧n = 2m} is non-regular. Explain each step of your reasoning with precision.

Simplify the following logic formulas. Use a truth table to show that your simplification is correct. (a.) (p∧q)∨q(p∧q)∨q (b.) p→(p→q)p→(p→q) (c.) ¬(¬p∧¬q)¬(¬p∧¬q) "Simplify" means: find the simplest/smallest formula that means the same thing.

The topic is on math logic Let M1 be a Kripke model with W = {wO, w1} and R = {{wo, w1}}. Moreover, suppose l(o, A) = 0 and I(wi, A) = 1. In this model, DA → A is not valid in wo. In other words, V(wO, DA → A) = 0. Exercise. Please prove this fact. (Note: this means DA → A is not true in a non-reflexive frame)