Suppose the domain of the propositional function 𝑃(𝑥, 𝑦) consists ofpairs 𝑥 and 𝑦, where 𝑥 is −1, 0 or 1 and 𝑦 is 0 or 2. Write out the propositions usingdisjunctions and conjunctions:a) ¬∀𝑥𝑃(𝑥, 2) (1 POINT)b) ∀𝑦∃𝑥𝑃(𝑥, 𝑦)

Suppose the domain of the propositional function 𝑃(𝑥, 𝑦) consists of pairs 𝑥 and 𝑦, where 𝑥 is 2, or 5and 𝑦 is 1, 2, or 3. Write out these propositions using disjunctions and conjunctions.a) ∀𝑥∀𝑦𝑃 (𝑥, 𝑦) b) ∃𝑥∃𝑦𝑃 (𝑥, 𝑦) c) ∃𝑥∀𝑦𝑃 (𝑥, 𝑦) d) ∀𝑦∃𝑥𝑃 (𝑥, 𝑦)

Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. [4 marks] a)∃x(x2 =2) b)∃x(x2 =−1) c) ∀x (x2 + 2 ≥ 1) d) ∀x (x2 =x)

Let 𝑓𝑓(𝑖𝑖, 𝑗𝑗) = 𝑖𝑖 𝑗𝑗𝑗𝑗!(a) Calculate 𝑓𝑓(2,3).(b) Calculate ∑ 𝑓𝑓(2, 𝑗𝑗)2𝑗𝑗=0 , ∑ 𝑓𝑓(𝑖𝑖, 3)3𝑖𝑖=1 .(c) Calculate ∑ ∑ 𝑓𝑓(𝑖𝑖, 𝑗

Let 𝑝 and 𝑞 be the proposition variables denoting𝑝: It is below freezing.𝑞: It is snowing.Write the following propositions using variables, 𝑝 and 𝑞, and logical connectives.a. It is below freezing and snowingb. It is below freezing but not snowingc. It is not below freezing and it is not snowing.d. It is either snowing or below freezing (or both).e. If it is below freezing, it is also snowing.f. It is either below freezing or it is snowing, but it is not snowing if itis below freezing.g. That it is below freezing is necessary and sufficient for it to besnowing

Let 𝐶(𝑥, 𝑦) mean that student 𝑥 is enrolled in class 𝑦, where the domain for 𝑥 consists of all students inyour school and the domain for 𝑦 consists of all classes being given at your school. Express each of thesestatements by a simple English sentence.a) 𝐶(𝑅𝑎𝑛𝑑𝑦 𝐺𝑜𝑙𝑑𝑏𝑒𝑟𝑔, 𝐶𝑆 252)b) ∃𝑥𝐶(𝑥, 𝑀𝑎𝑡ℎ 695)c) ∃𝑦𝐶(𝐶𝑎𝑟𝑜𝑙 𝑆𝑖𝑡𝑒𝑎, 𝑦)d) ∃𝑥(𝐶(𝑥, 𝑀𝑎𝑡ℎ 222) ∧ 𝐶(𝑥, 𝐶𝑆 252))e) ∃𝑥∃𝑦∀𝑧((𝑥 ≠ 𝑦) ∧ (𝐶(𝑥, 𝑧) → 𝐶(𝑦, 𝑧)))f) ∃𝑥∃𝑦∀𝑧((𝑥 ≠ 𝑦) ∧ (𝐶(𝑥, 𝑧) ↔ 𝐶(𝑦, 𝑧)))