Kuis Struktur Diskrit
Sifat: Open
Rabu, 6 Oktober 2021 15:00 – 17:00 WIB
Instruksi:
1. Menyalakan kamera saat mengerjakan kuis.
2. Kuis dikerjakan dengan ditulis tangan.
3. Unggah jawaban kuis yang telah di foto/pindai dengan format NIM_Nama.pdf pada Ms. Teams.
4. Kerjakan soal-soal berikut. Saat pengerjaan soal boleh tidak berurutan.
The Foundations: Logic and Proofs
1. Show the Equivalence law
𝑝 ↔ 𝑞 ≡
(
𝑝 → 𝑞
)
∧
(
𝑞 → 𝑝
) by constructing a truth table for
each side of the implication and showing these two truth tables are equivalent. Then use
the Implication law to rewrite
𝑝 ↔ 𝑞 ≡
(
𝑝 → 𝑞
)
∧
(
𝑞 → 𝑝
) in terms of
∧
,
∨
, and
~
only.
2. Simplify the following logical expressions with the laws of logic.
a.
~(p ∧ q) ∧ (~p ∨ q) ∧ (~q ∨ q)
b. [p
∨ (q ∧ r)
].
∧
.(~p
∨ r
)
Sets, Relations, & Functions
3. Using the Venn diagram find the sets
A, B, A ∪ B, A ∩ B, A!, B!
, (
A ∪ B
)
!,
(
A ∩ B
)
!, A!∪ B,
and
A ∪ B!
,
A − B
,
B − A
.
4. The below directed graph represents a relation. Find the domain and range of the relation.
Then write the relation as a subset of the Cartesian product of the domain and range.
5. Let A = {1, 2, 3, 4, 5, 6} with the relation given by
R = {(1, 1), (1, 5), (5, 1), (5, 5), (2, 2), (2, 4), (2, 6)
(4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6), (3, 3)}
⊂ A. × .A
.
Show this relation is an equivalence relation. Then find the equivalence classes that
partition the set A.
