Discrete Mathematics Answers

In how many ways can the word DISCRETE be arranged?

Use natural deduction to derive the conclusion in each problem.


Use conditional proof or indirect proof as needed:

1.
(x)(Dx ⊃ x = a)

2.
(x)[Ex ⊃(Dx • x = e)]
/ (∃x)(Ex ⊃ a = e)

Please help me with this question.
A number of students prepared for an examination in physics, chemistry and mathematics. Out of this number, 15 took physics, 20 took chemistry and 23 took mathematics. 6 took both physics and mathematics and all those who took physics also took chemistry. One student fell ill and failed to write any of the papers.
A. Draw a venn diagram for the above information
B. Use the venn diagram to find
I. How many students took one of the papers
II. How many students? prepared for the exams?

Find the total number of ways in which 10 different cakes can be divided into two parcels each containing the same number ?

In a class, 20 students study agric, 21 study biology and those who study chemistry are 5 more than those who study agric. it is also known that 3 students study agric only and 4 students study only chemistry. Half as many students who study chemistry only study biology only and twice as many students who study exactly one of the subjects study 2 of the 3 subjects. If one third of the students who study biology study all there subjects and the number of students who study agric and chemistry only is equal to the number of students who do not study any of the 3 subjects. a) Illustrate the above information on a vane diagram b) How many students i) are in the class ii) study at least 2 subjects iii) study at most one subject iv) Do not study any of the subjects

In a class, 20 students study agric, 21 study biology and those who study chemistry are 5 more than those who study agric. it is also known that 3 students study agric only and 4 students study only chemistry. Half as many students who study chemistry only study biology only and twice as many students who study exactly one of the subjects study 2 of the 3 subjects.
If one third of the students who study biology study all there subjects and the number of students who study agric and chemistry only is equal to the number of students who do not study any of the 3 subjects.

a) Illustrate the above information on a vane diagram
b) How many students
i) are in the class
ii) study at least 2 subjects
iii) study at most one subject
iv) Do not study any of the subjects

Use strong induction to prove that every integer n > 5 can be written as n = 4x + 3y for
non-negative x, y ∈ Z.

Xn
i=1
i2
i = (n − 1)2n+1 + 2

Use properties of Boolean algebra to simplify the following Boolean expression (show-
ing all the steps):
(x + y') (x + y')'