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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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Math Logic Problems
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