Description

1. Define the sets

A = {1,2,3}                                        B = {{1},{2},{3}}

C = {1,2,3,{2},{3},{1,2,3}} D = {{3},{2},{1},{1,2},{1,2,3}}.

Discuss the validity of the following statements

(explain why some are true and why the others are not true).

(a) A = B	(d) A ∈ C	(g) B ⊂ D
(b) A ⊆ B	(e) A ⊂ D	(h) B ∈ D
(c) A ⊂ C	(f) C ⊂ D	(i) A ∈ D

2 Determine (and explain why) whether the relation R on the set of all dogs is reflexive, symmetric, antisymmetric and/or transitive, where (a,b) ∈ R if and only if a) a runs faster than b;

1. a and b have the same fur colour;
2. a ate from the same bowl as b.

3. Sets describing intervals of real numbers are expressed with brackets and endpoints: a square bracket [ ] if the endpoint is included, a round bracket ( ) if the endpoint is excluded. Set A = [0,2). Then A is the set of all real numbers from 0 to 2, including 1 but not including 2. Define also the sets B = (−5,0) and C = [1,3].

• Write as intervals the 3 possible pairwise intersections and the3 possible unions of sets A,B,C. Name the resulting sets as D,E,…. Do not use different letters to denote the same set.
• You have several sets now. Define a relation ρ to be “is a subset of” ⊆, on the set consisting of all sets you obtained. Write down the ordered pairs of this relation and draw the Hasse diagram of this partial ordering.
• Does the resulting relation define a lattice? (explain why yes orwhy no)
• What is the least upper bound and the greatest lower bound ofthe set {A,B,C}?

20 marks 4. Define the relation ρ on the set S = {a,b,c,d,e,f} by

ρ = {(a,a),(b,b),(c,c),(d,d),(f,f)(a,b),(a,c),(c,a),

(b,c),(c,b),(e,d),(d,f),(e,f),(f,e)}

• Draw the directed graph of this relation.
• Verify whether this is an equivalence relation. If not, which pairsneed to be added to ρ to make it an equivalence relation? Write down its equivalence classes.

5. Given the binary relations on the set A = {1,2,3,4} defined by:

ρ₁ = {(1,4),(2,1),(2,2),(3,3),(4,3)}

and ρ₂ = {(1,2),(1,3),(2,3),(3,3),(4,4)}

determine (construct the ordered pairs) of the composite relations:

• ρ₁ ◦ ρ₂
• ρ₂ ◦ ρ₁
• ρ₁ ◦ ρ₂ ◦ ρ₁

6. (i) Use the properties of determinants (page 72 Study Guide (SG)) first to simplify and then to evaluate the determinants of A and B

(ii) Using the definition of rank of a matrix (3.3.1 P 74 SG), evaluate rank(B).          10 marks

7. (Extensions for higher marks) Calculate the determinants of the following matrices, and then solve for x the equations Det(A) = 0, Det(B) = 0

0 x + 1 3

.

20 marks 8. Prove that points (x₁,y₁),(x₂,y₂),(x₃,y₃) are collinear if and only if

.