CMPS 260 Spring 2020
Homework #1: Sets, relations, proof methods
Due: 4pm, Thursday, February 13

1. Let S and T be sets. For each equation provided, indicate the most general (i.e., most inclusive/least restrictive) relationship between S and T that would guarantee that the equation is satisfied.

Hint: Consider the four basic cases: one of S or T is a subset of the other (which has two sub-cases) or neither is a subset of the other (which has two sub-cases, depending upon whether their intersection is empty).

(a) S  =  S ∪ T
(b) S  =  S ∩ T
(c) S ∪ T  =  S ∩ T
(d) (S ∪ T) − T  =  S

2. Using as a model the proof of DeMorgan(1) found on the relevant web page, provide a proof of DeMorgan(2):  S ∪ T   =   S ∩ T

3. Recall these definitions pertaining to a binary relation R on a set A:

For each binary relation described below on the left, indicate in the table to the right which among the listed properties it possesses. Each relation is to be understood as being on the set A = { a, b, c }.

R1: { (b,c), (c,b) }
R2: { (a,b), (b,c), (a,c), (c,c) }
R3:  ∅   (i.e., empty set)
R4: { (a,b), (c,a) }
R5: { (a,a), (b,b), (c,c), (b,a), (b,c) } 		      
Relation
Property R1 R2R3 R4R5
reflexive       
symmetric       
transitive       

4. Let A = { a,b,c } and let P = { reflexivity, symmetry, transitivity }. For each of the three properties in P, a binary relation on A either possesses that property or fails to possess it. There are eight possible combinations of possession/non-possession —corresponding to the eight subsets of P— all of which are achievable.

In the previous problem, five relations over A were given and, for each one, you were to have identified which subset of the properties in P it possessed. Hence, at least three of the eight subsets of P were not represented in that problem. Choose two of those three subsets of P, and, for each one, identify it and describe a binary relation over A possessing exactly the properties in that subset.

5. Let R be a binary relation on the set A (i.e., R ⊆ A×A). Give a convincing argument in support of the following statement:  If R is both transitive and symmetric, and for each a ∈ A there is at least one b for which (a,b) ∈ R, then R is also reflexive (and therefore an equivalence relation!).

6. Let S(n) = Σ1≤i≤n(4i−1). That is, S(n) = 3 + 7 + 11 + ... + (4n − 1). Prove by mathematical induction on n that S(n) = n(2n+1) for all n≥1. Use as a model the proof in Example 1 of the relevant web page.