Maths - Linear Algebra - Sets - Assessment Answer

Maths Assignment

Assignment Task

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 ? 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 ? on the set S = {a, b, c, d, e, f} by

? = {(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 equivalence classes.

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

?1 = {(1,2),(1,3),(2,2),(2,3),(4,4)}

and

?2 = {(2,4),(3,1),(3,2),(3,3),(4,3)}

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

a) ?21

(b) ?1 ? ?2

(c) ?2 ? ?1

(d) ?1 ? ?2 ? ?1