Question 3. Refer to the definitions of reflexive, symmetric, antisymmetric, and transitive relations.

If 𝑅R is reflexive, symmetric, antisymmetric, and/or transitive, prove.

If 𝑅R is not reflexive, symmetric, antisymmetric, and/or transitive, provide a counter-example.

Question 6. Prove by contradiction, or disprove, the following two statements. (a) 55 is irrational.

(b) For any non-empty binary relation 𝑅R on the set 𝐴={𝑎,𝑏,𝑐,𝑑}A={a,b,c,d}, if 𝑅R is reflexive, transitive, and antisymmetric, then 𝑅R is not symmetric. (To prove by contradiction, state the assumption clearly.)

Question 1. Let 𝐶(𝑥,𝑦)C(x,y) be the predicate "Student 𝑥x is a student of class 𝑦y", 𝑃(𝑥)P(x) be the predicate "Student 𝑥x visited Po Toi Island", 𝑇(𝑥)T(x) be the predicate "Student 𝑥x visited Tai O", where the domain for 𝑥x contains all students in the college and 𝑦y contains all classes in the college.

(a) Translate the logical expression ∃𝑥(𝐶(𝑥,"𝐺𝑒𝑜𝑔𝑟𝑎𝑝ℎ𝑦")∧𝑃(𝑥))∃x(C(x,"Geography")∧P(x)) into an English statement.

(b) Express the statement "There is exactly one student in the "Culture" class who did not visit Tai O" using a logical expression with quantifiers.

Question 2. (a) On a fictional island, all inhabitants are either knights or knaves, where knights always tell the truth and knaves always lie. Based on statements made by A, B, and C, determine their identities.

(b) Assume three types of people on the island: knights, knaves, and spies (who can lie or tell the truth). Determine the identities of P, Q, and R based on their statements, knowing that one is a knight, one is a knave, and one is a spy.

Question 4. Without using a Venn diagram, prove or disprove the statement 𝑃(𝑆−𝑇)=𝑃(𝑆)−𝑃(𝑇)P(S−T)=P(S)−P(T) for any non-empty sets 𝑆S and 𝑇T.

Question 5. Determine whether each of the following functions 𝑓1,𝑓2,𝑓3f1,f2,f3 is injective and/or surjective, where 𝑓1((𝑎,𝑏))=𝑎+𝑏f1((a,b))=a+b, 𝑓2((𝑎,𝑏))=𝑎2−5f2((a,b))=a2−5, and 𝑓3((𝑎,𝑏))=([𝑏]−[𝑏])+[𝑎]−[𝑎]f3((a,b))=([b]−[b])+[a]−[a].

These adjustments clarify the intent of the questions and provide a structured framework for addressing each part effectively.