1. Given individual propositions p and q, verify the following logical equivalences by constructing a truth table for each part (T and F represent

Please see below! This is for Discrete Structures.

1. Given individual propositions p and q, verify the following logical equivalences byconstructing a truth table for each part (T and F represent true and false respectively): (a) (10 points) 10 A (T V p) E p A T(b) (10 pointS) n00 as g) E (:0 A 9‘) V (op A w)(G) (10 points} v“? A nq) -> (1)5 20 A nq 2. Given any individual propositions p, q, and r, use the laws of propositional logic (listedin Table 1.5.1 of the textbook) to prove the following logical equivalences. (a) (10 pointS) :0 A (a V or) E (p A a) V vb? V r)(b) (5 points) TA (pV T) E F V (F GET) 3. Formulate the following sentences using (nested) quantified statements. In each part,you need to specify appropriate domain for any variable participating in the statements. (a) (5 points) The product of any two negative integers is greater than their summa—tion. (b) (10 points) For any given positive number, there is another positive numbergreater than it and another positive number less than it. 4. (20 points) Using the DeMorgan’s law, negate the quantified statements obtained inthe previous question. 5. (20 points) State the contrapositive, converse, and inverse of the following conditionalstatements: (a) If the gold price increases, then, the oil price decreases. (b) If you become vegetarian, then, you will lose weight.