# 3 Refer to the definitions of reflexive, symmetric, antisymmetric, and transitive relations. If R is reflexive, symmetric, antisymmetric and/or transitive, prove.

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 πΆ(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 βπ₯(πΆ(π₯,"πΊππππππβπ¦")β§π(π₯))βx(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(πβπ)=π(π)βπ(π)P(SβT)=P(S)βP(T) for any non-empty sets πS and πT.

Question 5. Determine whether each of the following functions π1,π2,π3f1,f2,f3 is injective and/or surjective, where π1((π,π))=π+πf1((a,b))=a+b, π2((π,π))=π2β5f2((a,b))=a2β5, and π3((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.