Consider the following two assertions. Assume x is a specific but unknown real number.a. x < 2 or it is false that 1 < x < 3.b. x≤ 1 or either x < 2 or x≥ 3.Define statement letters for each of the individual statements (e.g., x < 2 is a statement). Hint: three statement letters will suffice. Using those statement letters, rewrite a and b as two logical expressions. Construct a truth table to show that a and b are logically equivalent.

(1) Complete the truth table and determine whether or not ∼(p∨q)≡∼p∧∼q p q p∨q ∼(p∨q) ∼p ∼q ∼p∧∼q T T           T F           F T           F F           (2) Are the two statement equivalent? A. Yes, the columns are identical. B. No, the rows are not identical.  C. No, the columns are not identical.  D. Yes, the rows are identical.

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.

1a.Suppose P is a false statement. Is it ever possible for P ⇒ Q to be false? Explain your answer. 1B.Suppose P is a true statement and that (P ∧ Q) ∨ ¬P is false. What is the “truth value” of Q? (No work needbe included with this question).  1C.Write down a non-statement, and explain why it is not a statement.

The next two questions refer to the following belief network with random variablesF0,F1,…,F4 which only take the values 0 and 1.   Which of the following statements are true about this belief network (1), (3) and (4) only :(1) F0 and F3 are independent given F1,F2 ;(2) F0 and F2 are independent;(3) P(F0∣F1,F2)=P(F0∣F1,F2,F3,F4) ;(4) F0 and F4 are independent. In general, to define the joint  probability distribution P(F0,…,F4) one requires up to 25−1 entries in the table of probabilities. For the belief network above, one can compute the joint probability distribution P(F0,…,F4) using the following number of conditional probabilities

Show if the statements are true or false and provide a brief explaination as to why. If A ∈ B and B ∈ C, is it true that A ∈ C? If A ∈ B and B ∈ C, then is it impossible that A ∈ C? If A ∈ B, then is it impossible for A to be contained in B?

Construct a truth table for (p ∨ ¬ q) ∨ (¬ p ∧ q) Use the truth table that you constructed in part 1 to determine the truth value of (p ∨¬q) ∨ (¬ p ∧ q), given that p is true and q is false. Determine whether the given statement is a tautology, contradiction, or contingency. p V (~p V q) ~ (p ∧ q) ~p V ~q

Propositions and quantifiers Let x, y ∈ {2, 3, 4}. For the following propositions provide aproof or counterexample to the statements below:•P0(x, y): x < y•P1(x, y): both x and y are even•P2(x, y): x ×y > 7•P3(x, y): x is even and x + y is a multiple of x.•P4(x, y): x ̸= y (a) for all propositions P in P0, . . . , P4,[∃x ∀y P ] →[∀y ∃x P ] (b) for all propositions P in P0, . . . , P4,[∀x ∃y P ] →[∃y ∀x P ]

5. Suppose X and Y are both TRUE, A and B are both FALSE, P and Q are unknown, which of the following can have the identifiable truth value? a. (P ∨ X) → B s. ( X ∧ Y) → P d. (P→X) ∧ Q f. (A ∨ P) ⇔ X

3. Express the following logical statements in terms of the Contrapositive, Converse, and Inverse,respectively.(a) You are an adult provided that you are above 18 years of age.(b) It is daylight whenever the sun is shining.(c) 3 × 2 = 6 only if 2 × 3 = 6.