Discrete Mathematics Answers

Calculate (by hand) the appropriate truth table to prove or disprove the following:
(p → q) ∧ (¬p → r) ⇒ (q ∨ r)
If it is invalid, give a counterexample; otherwise, explain why it is valid.

Let p and q bethepropositions“Theelectionisdecided” and“Thevoteshavebeencounted,”respectively.Express each of these compound propositions as an English sentence

1.In class of 40 students, 38 offer maths, 24 offer English. Each of the students offer at least one of the two subjects. How many students offer both subjects?

2. If A={ 3 ,4 ,5 ,9 and B={1,2,6,9,7}. Find (I) A–B (ii)B n A.

5.Let f be a function of A into B. If every member of B appears as the image of at least one element of A, then we say the function f is
a.surjective functions
b.constant function
c.injective functions
d.identity functions

6.Let f be a function of A into B, and let \\(b\\in B\\). Then \\(f^{-1}(b)=\\left \\{ x:x\\in A;f(x)=b \\right }\\)\n defines…
a.injective functions
b.constant function
c.inverse function
d.constant fnction

Give a direct proof, as well as a proof by contradiction, of the following statement:
‘ B A ∩ B ⊆ A ∪ for any two sets A and B .’

Let f : A → B and let X,Y be subsets of the domain A. For any Z ⊆ A, define the image of Z under f to be the set f[Z] = {b ∈ B|∃z ∈ Z(f(z) = b)}.

a. Show that f[X ∪Y] = f[X]∪f[Y].

b. Give an example of a function f and subsets X,Y of its domain to show that it is not always true that f[X ∩Y] = f[X]∩f[Y].

how many 4-letter words with or without meaning ,can be formed out of the letters of the word, "LOGARITHMS" ,if repetition is not allowed?

At the beginning of the first day (day 1) after grape harvesting is completed, a grape
grower has 8000 kg of grapes in storage. At the end of day n, for n = 1, 2, . . . , the grape grower
sells 250n/(n + 1) kg of their stored grapes at the local market at the price of $1.50 per kg.
During each day the stored grapes dry out a little so that their weight decreases by 2%.
Let wn be the weight (in kg) of the stored grapes at the beginning of day n for n ≥ 1.
(a) Find a recursive definition for wn. (You may find it helpful to draw a timeline.)
(b) Find the value of wn for n = 1, 2, 3.
(c) Let rn be the total revenue (in dollars) earned from the stored grapes from the beginning of
day 1 up to the beginning of day n for n ≥ 1.
Write a MATLAB program to compute wn and rn for n = 1, 2, . . . , num where num is entered
by the user, and display the values in three columns: n, wn, rn with appropriate headings.
Run the program for num = 20. (Use format bank.)

a) Show that (p → q) ∧ (p → r) and p → (q ∧ r) are logically equivalent.

Consider the sequences (rn) and (sn) defined recursively by
r0 = 1, s0 = 0, and rn+1 = rn/2, sn+1 = sn + rn+1 for n ≥ 0.
(a) What are the formulas for the nth terms rn and sn of these sequences?
(b) What is the floating point binary representation of sn?
(c) Write a MATLAB program which generates the sequences (rn) and (sn) recursively, and
run it on your computer until the computed values satisfy sn+1 = sn. To do this you can
use a for loop with a break statement. Display the sequence of values of n and sn in two
columns.
(d) What does this tell you about the storage of real numbers in your computer (assuming it
uses binary representation)?