Proof exercises, how to solve these? (discrete mathematics)
Question 2: A is a subset of B. So, N(A) <= N(B) . For any set S, P(S) = N(S)/N(Ω) . So, N(A)/N(Ω) <= N(B)/N(Ω). Or, P(A) <= P(B)(I suppose the part you are trying to prove here has a mistake, and you mean P(A) <= P(B) because the other thing makes no sense (since P(A) and P(B) are numbers.)Question 3: A - B contains everything that is included in A but not in B. Which would be A, minus the common elements of A and B. So (A, minus (A minus the common elements of A and B) = (the common elements of A and B) And there you have it.Cheers, happu learning!
Discrete Mathematics Help?
Let's start with some definitions. A relation R is: Reflexive if for every a, (a,a)∈R. Symmetric if (a,b)∈R implies (b,a)∈R. Anti-Symmetric if (a,b)∈R and (b,a)∈R implies a = b Transitive if (a,b)∈R and (b,c)∈R implies (a,c)∈R and you have given the definition of Asymmetric in your question. Now if you graph a relation using vertices and edges, then a is related to b (or aRb, or (a,b)∈R) if there is an arrow going from a to b. Now we can answer your questions. Graph #26 is: - reflexive. Graphically, if every vertex has a self-loop then it's reflexive. e.g. there is an edge from a to a, so (a,b)∈R. - not symmetric, because even though (c,a)∈R, we don't have (a,c)∈R. Graphically, if you have parallel edges going in opposite directions (or one undirected edge) between every pair of vertices, then it's symmetric. - not anti-symmetric, because we have (a,b)∈R and (b,a)∈R but a and b are two different vertices. Graphically, if a graph is anti-symmetric, there cannot be any self-loops or parallel edges in opposite directions between any two vertices. - not transitive, because we have (c,a)∈R and (a,b)∈R but (c,b) ∉R. Graphically you need that 'triangle' shape between every group of three vertices. i.e. If you can go from a to c via b, there should be an edge from a to b. - not asymmetric, because (a,b)∈R and (b,a)∈R. Graphically, if you have parallel edges in opposite directions between any two vertices, it's not asymmetric. I have given you a detailed explanation for the first graph. Here are the solutions for the rest. Graph #27: Reflexive: yes Symmetric: yes Anti-Symmetric: no Transitive: yes Asymmetric: no Graph #28: Reflexive: yes Symmetric: yes Anti-Symmetric: no Transitive: yes Asymmetric: no
Help me exercises about discrete mathematics (and,or,XOR)?
p XOR p is equivalent to (p ^ ¬p) v (¬p ^ p) p -> ¬q is equivalent to ¬p v ¬q which is equivalent to ¬(p ^ q)
Discrete Mathematics: Where can I find practice exercises/questions on Stable marriage problem?
Refer "Introduction to The Design & Analysis of Algorithms by Anany Levitin" for detailed explanation. Levitin has compiled good collection of algorithmic problems, it is a must have book on the desk.
Where are exercise solutions for Eric Lehman's Discrete Mathematics for CS available?
check the below links they contain answers for the assignment in the course of OCWAssignments | Mathematics for Computer Science | Electrical Engineering and Computer Science | MIT OpenCourseWareExams | Mathematics for Computer Science | Electrical Engineering and Computer Science | MIT OpenCourseWare
Can you help me with this mathematics exercise?
Let x be the side of square p = perimeter of square = 4 * x A = area of square = x * x Given: A = 2p x * x = 2 * (4 * x) x * x = 8x x^2 - 8x = 0 x(x - 8) = 0 x = 0, 8 x = 8 p = 4 * x p = 4 * 8 = 32 units
Are there any good free online courses for discrete mathematics?
Online Discrete Maths Study center at Carleton University. It has video lectures, lecture notes, and crucially, incremental exercises applying the lecture theory (the one thing that distinguishes from the 99.99 % of online material claiming to be a ‘course’)Discrete Mathematics Study Center
Where can I get the solutions to the exercise questions of Discrete Mathematics and its Applications, 7th edition - Rosen?
If you need to check your work, go for it. But keep the solutions far away from you until you've put in 9 hours per week, trying the problems on your own or with friends. Otherwise, you won't get the opportunity to actually develop the skills you need, and you'll be at a big disadvantage on exams and in subsequent classes.I once TAed an advanced class, in which the homework sets were extremely hard, and most of the students copied most of the solutions (sorry if this sounds made up, I feel it's inappropriate to give details). Despite homework counting for 20% of the class grade, about a third of the students with the highest HW scores got 'B-'s or worse. The correlation coefficient between homework and final exam scores was only 0.44.While I know it's valuable to be able to check your work, learning how to struggle with hard problems is one of the big opportunities provided by a discrete math class. By struggle I mean think hard about a problem, try different approaches, and hit a few dead ends over the course of half an hour (or sometimes more). You should learn, through practice, how to persevere and try things in an open ended setting, where many of the things you try may not work.Extremely few students have the discipline to allow themselves to struggle with a problem when they have access to a solutions manual. Consequently, they don't develop the skills needed to solve hard problems, on exams or in other classes.
What are some good books to learn discrete Mathematics?
(1) Discrete Mathematics and Application by Kenneth Rosen.This is a huge bulky book .Exercises are very easy and repeats a little . You can find good hints to the odd-numbered problems at the back of the book which is huge plus if you are self studying .(2)Elements of Discrete Mathematics by C.L. Liu . Short but nice read .(3) The art of Computer programming volume 1 by Donald Knuth . Very solid content . Only for mature readers .(4) Concrete Mathematics by Graham , Knuth and Patashnik . A highly regarded book in discrete mathematics . Contents can be used by any advance undergraduate . Problems are of varying level of difficult .Out of this four , if you are absolutely beginner i would suggest you to study (1) selectively and then go for (2) . After that you would be ready tackle (4) . After all of this read (3) if you are serious in discrete mathematics .After this if you are brave enough read thisCombinatorial problems and exercises by László LovászNot for the faint hearted . Problems are notoriously hard. I haven’t gone through it yet but from what i have heard it is very satisfying .