HW7_Sol.pdf - DISCRETE MATHEMATICS – CH7 Homework7 7.1 10 If A ={w x y z determine the number of relations on A that are(a reflexive(b symmetric(c

HW7_Sol.pdf - DISCRETE MATHEMATICS – CH7 Homework7 7.1 10...

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1 DISCRETE MATHEMATICS CH7 Homework7 7.1 10. If A = { w , x , y , z }, determine the number of relations on A that are (a) reflexive; (b) symmetric; (c) reflexive and symmetric; (d) reflexive and contain ( x , y ); (e) symmetric and contain ( x , y ); (f) antisymmetric; (g) antisymmetric and contain ( x , y ); (h) symmetric and antisymmetric; and (i) reflexive, symmetric, and antisymmetric. (10 pts) a. Reflexive 2 (𝑛 2 −𝑛) = 2 (16−4) = 2 12 b. Symmetric 2 𝑛 2 (𝑛 2 −𝑛)∗1/2 = 2 4 2 6 = 2 10 c. Reflexive and symmetric 2 (𝑛 2 −𝑛)∗1/2 = 2 6 d. Reflexive and contain (x,y) 2 12−1 = 2 11 e. Symmetric and contain (x,y) 2 𝑛 2 (𝑛 2 −𝑛)∗ 1 2 −1 = 2 4 2 5 = 2 9 f. Anti-symmetric 2 𝑛 3 (𝑛 2 −𝑛)∗1/2 = 2 4 3 6 g. Anti-symmetric and contain (x,y) 2 𝑛 3 (𝑛 2 −𝑛)∗ 1 2 −1 = 2 4 3 5 h. Symmetric and anti-symmetric {(1,1),(2,2),(3,3),(4,4)} Relation either include or exclude each of these pairs, so 2 4 i. Reflexive, symmetric and anti-symmetric : only 1 {(1,1),(2,2),(3,3),(4,4)} 7.3 (10 pts)
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2 7.4 12. Let A = { v , w , x , y , z }. Determine the number of relations on A that are (a) reflexive and symmetric; (b) equivalence relations; (c) reflexive and symmetric but not transitive;
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Unformatted text preview: (d) equivalence relations that determine exactly two equivalence classes; (e) equivalence relations where w ∈ [ x ]; (f) equivequivalence relations where v , w ∈ [ x ]; (g) equivalence relations where w ∈ [ x ] and y ∈ [ z ]; and (h) equivalence relations where w ∈ [ x ], y ∈ [ z ], and [ x ] ≠ [ z ]. (10 pts) a. 2 10 = 1024 b. ∑ S(5,i) 5 𝑖=1 = 1 + 15 + 25 + 10 + 1 = 52 c. 1024 − 52 = 972 d. 𝑆 (5,2) = 15 e. ∑ S(4,i) 4 𝑖=1 = 1 + 7 + 6 + 1 = 15 f. ∑ S(3,i) 3 𝑖=1 = 1 + 3 + 1 = 5 g. ∑ S(3,i) 3 𝑖=1 = 1 + 3 + 1 = 5 h. ( ∑ S(3,i) 3 𝑖=1 ) − ( ∑ S(2,i) 2 𝑖=1 ) = 3 3 Advanced assignment (30 pts)  Design a problem that can be solved by two different FSM with different number of states.  Use the minimization process to reduce the bigger one. Note:  FSMs in textbook will be scored 0.  The most similar FSM will be scored 0. Ans {0,2,4} , {1,3} {2,4} , {1,3} , {0} A= {0} , B{2,4} , C = {1,3}...
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