| 这是math510 离散数学课程的一份作业代写案例 Logical proof is a formal way of convincing that some statements are correct based on given facts. Let $P$ and $Q$ be two wff’s, predicates, or quantified predicates. Starting from $P$, if we can find a sequence of applications of laws of logic, rules of inference, or tautologies to arrive at $Q$, step by step, we say there is a proof for the theorem $$ P \Longrightarrow Q . $$ In theorem $P \Rightarrow Q, P$ is called the premise, and $Q$ is called a logical conclusion of $P$. The sequence of these steps is called a logical proof of $P \Rightarrow Q$. A logical proof is usually represented in a table like the following one. |
Consider
Premises: If Claghorn has wide support, then he’ll be asked to run for the senate. If Claghorn yells “Eureka” in Iowa, he will not be asked to run for the senate. Claghorn yells “Eureka” in Iowa.
Conclusion: Claghorn does not have wide support.
Determine whether the conclusion follows logically from the premises. Explain by representing the statements symbolically and using rules of inference.
Simplify
$$
(p \wedge(\neg r \vee q \vee \neg q)) \vee((r \vee t \vee \neg r) \wedge \neg q) .
$$
Let $D_{x}:=\mathbf{R}$. Which of the following are predicates?
(1) $x^{2}+1<0$
(2) $x$ is odd
(3) $\left(x^{2}-1\right) /(x+1)$
(4) $1+2=3$
(5) $x \in \mathbf{N}$
(6) $\sin ^{2} x+\cos ^{2} x$
直接用集合的包含关系做基本的计算即可验证
Let $P$ and $Q$ be two predicates, and let their truth sets be denoted as $T_{P}, T_{Q}$, and falsity sets be denoted as $F_{P}, F_{Q}$. Prove the following identities.
- $T_{P} \cap T_{Q}=T_{P \wedge Q}$
- $T_{P} \cup T_{Q}=T_{P \vee Q}$
- $F_{P} \cap F_{Q}=F_{P \vee Q}$
- $F_{P} \cup F_{Q}=F_{P \wedge Q}$
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离散数学代写
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Homeworks
Note: Homework numbers refer to the Fifth Edition of Brualdi.
HW1. 1.8. 4a(also compute f(13)),12,16,17,34,42. 2.7. 1,4b,5b,6. (Fri 1/29).
HW2. 2.7. 7,9,12,13b,14,20,23,24,25,28,31,57. (Fri Feb 5).
HW3. 2.7. 19b,21,37,38,42,43,45,46,51,63. (Fri Feb 12).
HW4. 3.4. 5,7,9,10,14,16,17,19b,20,23. (Fri Feb 19).
HW5. Homework 5 (Due Fri Feb 26).
HW6. 5.7. 10,11,20,22,25,27,29,37,38,40. (Due Fri Mar 4).
HW7. 5.7. 36,47. 6.7. 2,3,5,6,8,9,13,17. (Due Fri Mar 11).
HW8. 6.7. 14,19,21,24ac,26,27,28,29,31,32. (Due Fri Mar 25).
HW9. 7.7. 1d,2,3c,5,9,11a,32,33,38e,39,40. (Due Fri Mar 32).
HW10. 7.7. 13bce, 14bde,15,18,34,36,47,52. (Due Fri Apr 8).
HW11. 7.7. 26,28,48d,51. 11.8. 2,6,9,11,12. (Due Fri April 15).
HW12. 11.8. 14,15,17,21,29,30,33,37,49ab. (Due Mon April 25).
HW13. 11.8. 20,44,47,54,55,56,62,64. (Due Fri April 29).
HW14. 13.4. 29,30. 9.4.1*,2*,6*,7*,19,25 (women choose). Bonus questions: 12.7. 5,26,27. (Due Fri May 6).
For * questions use bipartite graphs & max-matchings instead of SDRs.
“Suppose aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack.”
Paul Erdős – as quoted in “Ramsey Theory” by Ronald L. Graham and Joel H. Spencer, in Scientific American (July 1990), p. 112-117
