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

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

If π R 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 π R on the set π΄={π,π,π,π}A={a,b,c,d}, if π R is reflexive, transitive, and antisymmetric, then π R is not symmetric. (To prove by contradiction, state the assumption clearly.)

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

(a) Translate the logical expression βπ₯(πΆ(π₯,"πΊππππππβπ¦")β§π(π₯))βx(C(x,"Geography")β§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 π(πβπ)=π(π)βπ(π)P(SβT)=P(S)βP(T) for any non-empty sets πS and πT.

**Question 5.** Determine whether each of the following functions π1,π2,π3f1,f2,f3 is injective and/or surjective, where π1((π,π))=π+πf1((a,b))=a+b, π2((π,π))=π2β5f2((a,b))=a2β5, and π3((π,π))=([π]β[π])+[π]β[π]f3((a,b))=([b]β[b])+[a]β[a].

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