SIT292 : Linear Algebra : Maths – Engineering Assignment Questions : 1. Let us have the following sets A = {u, v, w} B = {{u}, {v}, {w}} C = {u, v, w, {v}, {u, v, w}} D = {{w}, {v}, {u}, {u, v, w}}. 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 whether the relation R on the set of all people is reflexive, symmetric, antisymmetric and/or transitive, where (a, b) ∈ R if and only if a) a runs faster than b; b) a and b graduated on the same day; c) a and b have a common cousin. 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. Let A = [2, 6). Then A is the set of all real numbers from 2 to 6, including 2 but not including 6. Also let us have the sets B = (1, 4) and C = [0, 2]. (a) Write as intervals the 3 possible pairwise intersections and the 3 possible unions of sets A, B, C. Name the resulting sets as D, E, … (b) You have 9 sets now. Define a relation ρ to be “is a subset of” ⊆, on the set consisting of all 9 intervals you get. Write down the ordered pairs of this relation and draw the Hasse diagram of this partial ordering. 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),(e, e),(f, f)(a, b),(b, a),(a, c),(c, a), (b, c),(c, b),(d, e),(e, d),(d, f),(f, d),(e, f),(f, e)} (a) Draw the directed graph of this relation. (b) Verify that this is an equivalence relation. Write down its equiv- alence classes. 5. Given the binary relations on the set A = {1, 2, 3, 4} defined by: ρ1 = {(1, 2),(1, 3),(2, 2),(2, 3),(4, 4)} and ρ2 = {(2, 4),(3, 1),(3, 2),(3, 3),(4, 3)} determine (construct the ordered pairs) of the composite relations: (a) ρ21 (b) ρ1 ◦ ρ2 (c) ρ2 ◦ ρ1 (d) ρ1 ◦ ρ2 ◦ ρ1
SIT292 :
Questions :
1. Let us have the following sets
A = {u, v, w} B = {{u}, {v}, {w}}
C = {u, v, w, {v}, {u, v, w}} D = {{w}, {v}, {u}, {u, v, w}}.
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 whether the relation R on the set of all people is reflexive, symmetric, antisymmetric and/or transitive, where (a, b) ∈ R if and only if
a) a runs faster than b;
b) a and b graduated on the same day;
c) a and b have a common cousin.
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. Let A = [2, 6). Then A is the set of all real numbers from 2 to 6, including 2 but not including 6. Also let us have the sets B = (1, 4) and C = [0, 2].
(a) Write as intervals the 3 possible pairwise intersections and the 3 possible unions of sets A, B, C. Name the resulting sets as D, E, …
(b) You have 9 sets now. Define a relation ρ to be “is a subset of” ⊆, on the set consisting of all 9 intervals you get. Write down the ordered pairs of this relation and draw the Hasse diagram of this partial ordering.
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),(e, e),(f, f)(a, b),(b, a),(a, c),(c, a), (b, c),(c, b),(d, e),(e, d),(d, f),(f, d),(e, f),(f, e)}
(a) Draw the directed graph of this relation.
(b) Verify that this is an equivalence relation. Write down its equiv- alence classes.
5. Given the binary relations on the set A = {1, 2, 3, 4} defined by:
ρ1 = {(1, 2),(1, 3),(2, 2),(2, 3),(4, 4)} and
ρ2 = {(2, 4),(3, 1),(3, 2),(3, 3),(4, 3)}
determine (construct the ordered pairs) of the composite relations:
(a) ρ21
(b) ρ1 ◦ ρ2
(c) ρ2 ◦ ρ1
(d) ρ1 ◦ ρ2 ◦ ρ1
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