Are the following statement forms logically equivalent: p ∨ q → p and p ∨ (∼ p ∧ q)

2. Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. If true, give a brief explanation for why. If false, give a counterexample (i.e. a set of values x,y that make the statement false. You must also explain why they make the statement false.) a) yvx((y > x²) → y > x) b) Vy!x(√√ 0)) d) x(x < 0⇒x² ≥y) e) \xy(x² = y² ↔ y³ = x³)

Set up truth tables for the statementsP ⇒ (P ⇒ Q), (P ⇒ Q) ⇒ Q, P ∧ (∼ Q), (∼ P) ∨ (∼ Q)Are any of these statements equivalent?

For each of the following statements about regular expressions α, β and languages A, B, state whether they are true or false. Provide a one-sentence justification for each answer.

the quetion is below, im finding it hard to undertsand the topic

Exhibit the structure of the following statements by transforming them into a first-order formula,indicating the interpretation of predicates and the domain. For example the statement There isno greatest integer can be transformed into ¬∃x, ∀y, P(x, y) where P(x, y) is the predicate x ≥ y,the domain being the integers.Do not try to prove them (it may not be possible!).1. Every integer can be written as the sum of 2 squares.2. Every positive real number has a square root.3. The cosine function has zeroes.4. The cosine function has at least two distinct zeroes.5. There is a neutral element1for multiplication in real numbers.6. Every odd square can be written as the sum of three odd numbers.

Question 2 Consider the correctness statement x == x { x = x + 1; } P where x has type integer. The correctness statement is valid when P is the assertion: Ox>0 Ox 1 Ox< 1 false The correctness statement is invalid in all above cases.