Discrete Mathematics Answers

Construct a truth tabal for each of these compound proposition

Find the inverse of 35 modulo 11 by using extended Euclidean Algorithm

Use properties of Boolean algebra to simplify the following Boolean expression (showing all the steps):

(x + y′)(x′ + y′)′

Show that the hexadecimal expansion of a positive integer can be obtained from its binary expansion by grouping together blocks of four binary digits, adding initial zeros if necessary, and translating each block of four binary digits into a single hexadecimal digit.

Using a truth table, prove or disprove the following:

~[(~p∧q)↔ r]≡ ~(p∨~q)↔ ~r

Show that whether x5 + 10x3 + x + 1 is O(x4) or not?

Determine the cardinality of each of the sets, A, B, and C defined below, and prove the cardinality of any set that you claim is countably infinite.

A is the set of negative odd integers

B is the set of positive integers less than 1000

C is the set of positive rational numbers with numerator equal to 1

1. Write the forpseudocode  an algorithm that determines if a function is injective, assuming that you can iterate through all the elements in the domain and the co-domain in a finite number of steps.
   
2. Recall the insertion sort algorithm discussed in class (pseudocode shown below).  Find the big-O of the number of comparisons made in the execution of insertion sort on the input list of numbers:  k, k-1, k-2, ..., 2, 1.  Explain your reasoning.

ALGORITHM 5 The Insertion Sort.

procedure insertion sort(a1, a2, ..., : real numbers with n > 2) for j = 2 ton i=1 while aj > a i=i+1 m = 0; for k:= 0 to j-i-1 aj-k := 0;-k-1 dim {a1, ..., An is in increasing order)

5.

1. Find the O(g) estimate for each of the following functions. The order of g should be as small as possible, and g should be as simple as possible.
2. a) nn100+nn(n!)+n1.2n

b) (n5+100)(32 log n - 6) + log n (n5 + 2n4 log n)

c) (n6+1.5n)(1.1n+n5)

Determine the cardinality of each of the sets, A, B, and C defined below, and prove the cardinality of any set that you claim is countably infinite.

A is the set of negative odd integers

B is the set of positive integers less than 1000

C is the set of positive rational numbers with numerator equal to 1

Obtain the sum – of – products and product - of - sum canonical form for 𝑥1⨁(𝑥2 ⋆ 𝑥3 ′ )