Discrete Mathematics Answers

Consider the statement form

(P \downarrow Q) \downarrow R

Now, find a restricted statement form logically equivalent to it, in

a) Disjunctive normal form (DNF).

b) Conjunctive normal form (CNF).

a) Prove that the set { \downarrow } is an adequate set of connectives.

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b) Consider the set

F = {y | y = ax³ + b},

a, b being constants such that a != 0 and x   \mathbb {R}.
Is F \approx \mathbb{R}? If so, prove it. If not, explain in details why it is not the case.

a) Consider the function, bin, that converts an integer to a binary digit as:

bin : Z 6{-1, 0, 1}, bin(x) = x congruence modulo 2.

Now, define the logical NOT function as:

NOT: Z"\\to" {0, 1}, NOT(x)="\\begin{Bmatrix}\n 0, if x != 0\\\\\n 1, if x=0 \n\\end{Bmatrix}"

i) Write C++ code for the logical NOT function specified above.

ii) Using the function defined in i) above, write in C++ the polymorphic logical exor

function:

int exor(int x, int y):- it returns x r y

int exor(int * bits, int N):- it returns the exor of the N binary integers, bits.

b) As a result of a) above, write in C++ the polymorphic logical not exor function,

nexor:

int nexor(int x, int y):- it returns the negation of x r y

int nexor(int * bits, int N):- it returns the nexor of the N binary integers, bits

a) let us define the following:

1. A “vowel group” is a vowel, v "\\in" {‘a’, ‘e’, ‘i’, ‘o’, ‘u’}, or a concatenation of

“two” vowels.

2. A “vowel syllable” is a vowel group.

3. A “consonant syllable” is a “one”, or “two” or at most “three” consonants, c "\\in"

{‘a’, ‘b’, ‘c’, ..., ‘z’} - {‘a’, ‘e’, ‘i’, ‘o’, ‘u’} - {‘d’, ‘v’, ‘w’, ‘x’, ‘z’}, followed by

a vowel group.

Now, a Sesotho word can be described as follows:

1. A vowel syllable is a Sesotho word.

2. A consonant syllable is a Sesotho word.

3. A Sesotho word can begin with a vowel syllable, and thereafter it could be a

mixture of vowel syllables and/or consonant syllables (with at most “two”

consonants).

4. A Sesotho word can start with a consonant syllable, and thereafter it could be a

mixture of vowel syllables and/or consonant syllables.

Draw a finite state automaton (FSA) that can generate/accept Sesotho words as

defined above.

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1.construct the truth table for the following compound propositions. Assume all variables denote propositions.A. (pVq) ^ [(~p^q)]B. ~(p—>(q—>(p^q)))C. (p<—>q) V ((~p)—>q)D. (p<—>q) V ((~p)—>(~r))E. (p—>(q—>r))—>((p^q)—>r)

Translate the given statements into propositional logic using the propositions provided

You can upgrade your operating system only if you have a 32-bit processor running at 1 GHz or faster, at least 1 GB

RAM, and 16 GB free hard disk space, or a 64- bit processor running at 2 GHz or faster, at least 2 GB RAM, and at least

32 GB free hard disk space. Express you answer in terms of u: “You can upgrade your operating system,” b32: “You have

a 32-bit processor,” b64: “You have a 64-bit processor,” g1: “Your processor runs at 1 GHz or faster,” g2: “Your processor

runs at 2 GHz or faster,” r1: “Your processor has at least 1 GB RAM,” r2: “Your processor has at least 2 GB RAM,” h16:

“You have at least 16 GB free hard disk space,” and h32: “You have at least 32 GB free hard disk space.”