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MATH1081 - Discrete Mathematics Assignment

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Added on  2021-05-29

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The article provides solutions for MATH1081 Discrete Mathematics Assignment. It covers topics like subset relations, partial order, and divisibility for sets and relations.

MATH1081 - Discrete Mathematics Assignment

   Added on 2021-05-29

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MATH1081 - DISCRETE MATHEMATICS ASSIGNMENT2020 T11.Consider the following sets:S = {180n + 33 | n ∈ Z} ,T = {18n − 3 | n ∈ Z} ,andU = {20n + 13 | n ∈ Z} .(a) Show that S is a proper subset of T .Let s ∈ S.By the definition of S, there exists a k ∈ Z such thats = 180k + 33,= 180k + 36 − 3,= 18(10k + 2) − 3,where 10k + 36 is an integer.Hence, s ∈ T and so S ⊆ T .Now, observe that 15 ∈ T since 18(1) − 3 = 15 and 1 ∈ Z.However, if 180n + 33 = 15,180n = −18,n = −110.Since n /∈Z, 15 /∈ S and hence T * S.Therefore, S is a proper subset of T .(b) Show that S is a proper subset of U .Let s ∈ S.By the definition of S, there exists a k ∈ Z such thats = 180k + 33,= 180k + 20 + 13,= 20(9k + 1) + 13,where 20k + 13 is an integer.Hence, s ∈ U and so S ⊆ U.Now, observe that 13 ∈ U since 20(0) + 13 = 13 and 0 ∈ Z.However, if 180n + 33 = 13,180n = −20,n = −19.Since n /∈Z, 13 /∈ S and hence U * S.Therefore, S is a proper subset of U .1
MATH1081 - Discrete Mathematics Assignment_1

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