160.212 Discrete Mathematics Assignment 2
School of Mathematical and Computational Sciences Mathematics
160.212 Discrete Mathematics
Assignment 2 Semester One, 2024
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Let N be the set of all divisors of 60, and let
P ={a∈N|4≤a≤30}.Let ≼ be the partial order defined by a ≼ b if and only a divides b.
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(a) Draw the Hasse diagram of (P, ≼).
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(b) For every pair (a,b) of numbers in P evaluate the least upper bound, lub(a,b), or say it does not exist (DNE). Present the values in the form of a table.
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(c) Use your answer to (b) to explain why (P, ≼) is not a lattice.
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Consider the function f : R2 → R2 defined by
f(x,y)=y+1−7x2, 3 x. 5 10
This is the H ́enon map used widely as a paradigm for chaos.
(a) Is f an injection (one-to-one)? Justify your answer. (b) Is f a surjection (onto)? Justify your answer.
(c) Deriveaformulaforf◦f.
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Let ∗ be the binary operation on R defined by
x ∗ y = min(x, y).(a) Prove that ∗ is commutative and associative.
(b) Does ∗ have an identity element? Justify your answer.(c) Is (R, ∗) a monoid? Justify your answer.
4. The multiplicative group modulo 30 is (M30,×30), where M30 is the set of all integers
1 ≤ n < 30 with gcd(n, 30) = 1, and ×30 is multiplication modulo 30.
(a) Determine the order of this group. (b) Show that {1, 11} is a subgroup.
(c) Compute ⟨7⟩ (the cyclic subgroup generated by 7). 5. Consider the permutation group S5, and let f, g ∈ S5 be
f=1 2 3 4 5,
43251 35142
(a) Evaluate f ◦ g. (b) Evaluate g−1.
(c) Findh∈S5 suchthatf=g◦h.
2
g=1 2 3 4 5.
