Exercise 4: (4.5 POINTS) Determine the truth values of the following propositions. Explain.a) ∅ ∈ {1}b) {1} ∈ {1}c) {1} ⊂ {1}d) {1} ∈ {1, {1}}e) {1} ⊆ {1, {1}}f) {{1}} ∈ 𝒫({1, {1}})

Determine whether each of these statements is true or false.a) 0 ∈ ∅b) ∅ ∈ {0}c) {0} ⊂ ∅d) ∅ ⊂ {0}e) {0} ∈ {0}f) {0} ⊂ {0}g) {∅} ⊆ {∅}

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)

Exercise 2: (3 POINTS) 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)

U = {1, 2, {1}, {2}, {1, 2}}      A = {1, 2, {1}}       B = {{1}, {1, 2}}     C = {2, {1}, {2}}.Which one of the following statements is valid if x ∉ B U C? (Hint: Determine U – (B U C).)a.x ∈ {1}.b.x ∈ ⊘.c.x ∈ {1, 2}.d.x ∈ B and x ∈ C.

Let A = {{⊘}, 1, {1}}.Which one of the following alternatives regarding A is FALSE?a.{{1}} ⊂ Ab.{⊘} ⊂ Ac.|Ƥ (A)| = 8d.{⊘} ∈ AClear my choice