COMP9020 Quiz 2, Test started Feb 21, 2024 6:00 PMtoFeb 28, 2024 6:00 PM

1.1

Let A be the set {b,a,n,a,n,a} and B be the set {p,i,n,e,a,p,p,l,e}.
Which of the following are subsets of A⊕B?
Select all that apply

- {b,p,i,e,p,p,l,e}
- {a,b,e,i,l,n,p}
- {b,p}
- {p,p,p,a,a,n,n,e,e}
- {}
- {a,n}

重复的元素视为同一元素即可。

因此，答案有：

- {b,p,i,e,p,p,l,e}
- {b,p}
- {}

1.2

Which of the following sets have exactly 6 elements?  (Intervals are over ℕ)
Select all that apply

- [0,1] × (1,6)
- (0,1) × [1,6]
- (0,2] × [1,4)
- [0,2] × (1,4)
- None of the above

• 选项1：

• 选项2：

• 选项3：

• 选项4：

因此，答案有：

- (0,2] × [1,4)
- [0,2] × (1,4)

1.3

Let Σ={a,b}. 
Let w ∈ Σ* be the word "abb",
let v ∈ Σ* be the word "ba".
What is length(vwv)?

1.4

Let Σ={c,u,p} and Ψ={s,a,u,c,e,r}
Which of the following words are in Σ* \ Ψ*?
Select all that apply

- ccc
- eee
- saucer
- λ
- cup
- ppp

要点：寻找能在集合Σ中组出来的词，并且这个词在集合Ψ中无法组出来。

因此，答案有：

- cup
- ppp

1.5

Which of the following statements are true for all sets A, B, C?
Select all that apply

- (A ∪ B)\C = (A\C) ∪ (B\C)
- (A⊕B)ᶜ = (Aᶜ) ⊕ (Bᶜ)
- (C\A)ᶜ = (Cᶜ) \ (Aᶜ)
- (A∪B)∩A = A∪(B∩A)
- C\(A ∪ B) = (C\A) ∪ (C\B)

• 选项1：

• 选项2：

• 选项3：

• 选项4：

• 选项5：

因此，答案有：

- (A ∪ B)\C = (A\C) ∪ (B\C)
- (A∪B)∩A = A∪(B∩A)

2.1

For all sets A and B, A∩B = A∪B  if and only if A=B.

• if:

if A=B, then A∩B = A∪B.

• only if:

if A∩B = A∪B, then A=B.

因此，答案为：

- True

2.2

For all sets A,B,C:
A×(B∩C) = (A×B) ∩ (A×C)

2.3

Which of the following sets has cardinality less than or equal to 6?
Select all that appy:

- {n∈ℤ : 6|n}
- {6%n : n∈ℕ and n>0}
- {n∈ℤ : n|6}
- {n%6 : n∈ℤ}

• 选项1：

在集合 $\{n \in Z: 6 \mid n\}$ 中，$n$ 代表了整数集合中所有6的倍数，因此是该集是无穷的。

• 选项2：

• 选项3：

• 选项4：

因此，答案有：

- {6%n : n∈ℕ and n>0}
- {n%6 : n∈ℤ}

2.4

Suppose A={0,1,2}
For how many sets B⊆ℕ is it the case that A×B = B×A?
Select one alternative:
- 0
- 1
- 2
- 3
- Infinitely many
- None of the above

假设集合B不为空集，且元素为：$\{b_1,b_2,b_3,...b_n\}$，那么：

若要保证 $A \times B = B \times A$ ，需要有：

因此，当 $B=\{0,1,2\}=A$ 时，有 $A \times B = B \times A$。

假设B为空集，显然：

因此，答案有：

- 2

2.5

Let Σ = {0,1}.
True or false:
For all languages X ⊆ Σ*: (XX)* = (X*)(X*)

反例：

单字符的字符串无法被$(XX)^*$表示。

因此，答案为：

- False