If you’re missing out on your usual support class on Tue 25 Apr, you can submit your assignment under the door of Daniel’s office (9Rnf/418) by 2:30pm Fri 28 Apr. Please clearly mark your support class day,
time and room on your assignment.
1. For each of the binary relations E, F and G on the set {a,b,c,d,e,f,g,h,i} pictured below, state whether the relation is reflexive, symmetric, antisymmetric and transitive. When a relation does not have one of these properties give an example of why not.
2. For those (if any) of E, F and G that are equivalence relations, state the equivalence classes. Answer only required.
3. For those (if any) of E, F and G that are partial order relations, are they total order relations? Fully justify your answer.
4. Let R be binary relation defined on P(N) by ARB if and only if |A∩B| ≤ 2. Is R reflexive? Is it symmetric? Is it antisymmetric? Is it transitive? Fully justify each of your answers.
5. Let S be equivalence relation defined on {x : x ∈ R and 0 ≤ x ≤ 5} defined by xSy if and only if dxe = dye. What are the equivalence classes of S? Answer only required.
Note: dqe is defined to be the smallest integer greater than or equal to q. You can think of it as “q rounded up”. You don’t need to prove that S is an equivalence relation.
6. Let T be the partial order relation defined on N × N by (a,b)T(c,d) if and only if a ≤ c and b ≤ d. Is T a total order relation? Fully justify your answer.
Note: You don’t need to prove that T is a partial order relation.
