## Transcribed Text

2 Sets, partial orders, and graphs
Problem 2.1. Let bea that closed under union that that ifAEMandBe then
AUB € M. Prove the following statement by mathematica induction
M, for all natural
Problem 2.2 Let and let L be A*, thati the set all strings over Let
y mean that string sel such that x$ For example aaa (aaabbb (because
(because there is no such thatbs aab). Prove that
is
partial
new graphs from already existing graphs.
Definition
vertices
Vcis
the
set of
tuples
two
This pretty abstract definition, so here an example: If G is the graph with vertices
b,c edges E thon tho vorticas in GC the set
(c,b)}.
Wecan draw the graphs and GC follows
(b,a) (c,a)
(b,c GC (c,b)
(a,c) (a,b)
Problem 2.3. Let bothe graph whore V. Specify the graph Gc.
Problem 2.4. Suppose that isa graph with vertices Find formula that expresses
the
number vertices ir the graph Gc Explain why this formula gives the number vertices :
GC-
Problem 2.5. Now let be the graph with verticos V=(a b,c}and edges (b,c})
Show that this countercxample tothe statement that says "for all sonnected then
GC sconnected.
3
Figures and functions
In this oblem we will work with figures that look follows Each figure consists of grid
with two columns and three rows, where each square either white or black
We think of white square as and ablack square as and figures can thu be represented
as sib strings The four figures 000000, 100001 010101, and 000111, respectively.
Definition
Let figures the form where a.b.c, d,e and f are elements in (0,1).
refer this figure by drawing thegrido writing abcdef When wedraw thegrid
we use white square for and black square for 1
Here are all the elements M. You may answer all the relevant subproblems by referring to
the bit strings
Definition
Let and H be the following functions from M to M.
Notice that are placeholders in this definition Onewaytothink about these
functions sthat Lis the identity, that rotates figure that reflects a figure vertically and
that reflects figure horizontally. Here are some examples Fhow the functions work:
Asa repetition, hereis the definition functions:
Definition
If
B
-C
are
functions,
then
function
defined
first applying
this,
that
is,
This
the
composition
Problem 3.1. of(V. H)(101000), (Ho R)(001010) and (V R)(010100).
Problem 3.2. Specify the function (R IV oH)).
Problem 3.3. Let IIR V.H). Prove that (G1,o)isa group.
Problem 3.4. Decide from the previous problem is an abelian group Justify your
answer.
Definition
Iffisa function, and fix) wesay that xis fixed point for f.
Problem 3.5. Find all elements that are fixed points for the function V.
Definition
Let be the relation on M. such thatx function f G, such that f(x)=y
Show that is an equivalence relation.
equivalence classo 101000?
Definition
function from M. M
Problem
3.8.
own
Problem 3.9. of D(110000), (D(110000) and D(D(D(110000))).
Problem 3.10. Let G2 D. and G:be the closure G2 under that is the smallest
set containing IID.H} that closed under Find out different from G2. Arethey
identical? Does contain more elements than G2? Ifso. which elements?

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