这个作业是完成离散数学相关的题目

UL21/120 Discrete Mathemaitcs

问题1集合论
(a)我使用清单方法描述以下集合:
A = {x:x∈Z并且0≤x
3 <100}
[2]
ii。使用集合构建器方法重写以下集合:
B = {−1,1/2,−1/3,1/4,−1/5,1/6,···}
[2]
(b)在对200名学生的调查中发现:150名学生参加了编程
(P),数学专业80名学生(M),艺术专业55名学生(A),60名学生
攻读数学和程序设计(M&P)课程,25名学生攻读美术
和数学(A&M),有40名学生修了艺术和程序设计(A&P),
15名学生修读艺术,数学和程序设计(A&M&P)。
一世。绘制维恩图以显示此信息。
ii。使用维恩图查找参加课程的学生人数
1.仅编程
2.仅两个模块
3.数学和程序设计而非艺术
[6]
(c)令A和B为通用集合U的两个子集。证明或反对
P(A∪B)= P(A)∪P(B)。
[4]
(d)令A和B为通用集合U的两个子集。证明:
A⊆B⇔A∪B = U
问题2的功能
(a)设f:R→Z.其中f(x)= b
X
2
C。
一世。求f(1)[1]
ii。什么是10的原像集[1]
iii。说出f(x)是否是内射(一对一),证明你的答案是正确的。 [2]
iv。说出f(x)是否是射影(上),证明您的答案是正确的。 [2]
(b)给定一个函数g:R→R由g(x)= 3x + 5定义。
一世。证明函数g是双射。 [2]
ii。找g
-1
[2]
(c)令f:Df→[0,=∞)是f(x)= ln(x + 1)的双射函数。
一世。查找此函数的域Df。 [1]
ii。找到反函数f
-1
。 [2]
iii。绘制函数f和f的曲线
-1
在同一张图中。 [2]
iv。您能对这两条曲线说些什么? [1]
(d)确定是否从Z×Z到
Z是一对一,对上或两者兼有。解释你的答案。
一世。 f(x,y)= x
2 + 1
ii。 g(x,y)= x + y + 2
Question 3 Propositional Logic
(a) Let p, q, r and s four propositions. Assuming that p and r are false and that
q and s are true, find the truth value of each of the following propositions:
i. ((p ∧ ¬q) → (q ∧ r)) → (s ∨ ¬q)
ii. ((p ∨ q) ∧ (q ∨ s)) → ((¬r ∨ p) ∧ (q ∨ s))
[2]
(b) Let p and q be two propositions defined as follows: p means ’A student can
take the algorithm module ’ whereas q means ’Student passes discrete
mathematics’
Express each of the three following compound propositions symbolically
by using p, q and appropriate logical symbols.
i. ‘A sufficient condition for a student to take the algorithm module is that
they pass discrete mathematics’.
ii. ‘A student can take the algorithm module only if they pass discrete
mathematics’.
iii. ‘ A student can takes the algorithm module if they pass discrete mathematics’.
iv. ‘ A student either passes discrete mathematics or can take the algorithm
module’
[4]
(c) Write in words and express symbolically in terms p and q, defined in (a),
the contrapositive, the converse and the inverse of the implication:
‘A student can take the algorithm module if they pass discrete mathematics
’ [6]
(d) Consider the following three propositions:
s means ’”Samir goes to the party ”
c means ’”Callum goes to the party”
j means ’”Jay goes to the party”.
Express each of the three following compound propositions symbolically
by using c, j, s and appropriate logical symbols.
i. “Samir goes to the party only if both Callum and ‘Jay aren’t going to
the party.
ii. “Either both Samir and Jay go to the party or Callum goes to the party,
but not both ”.
[4]
(e) A tautology is a proposition that is always true. Let p and q be two
propositions, show that (p → q) ⇔ (¬q → ¬p) is a tautology.


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