Discrete Mathematics Answers

Solve the following recurrence relations
i) Fn= Fn-1 +Fn-2 where a1=a2=1
ii) an=2an-1 - an-2 +2 where a1 = 1 and a2 = 5

1.Determine whether ¬(p∨(¬p∧q)) and ¬p∧¬q are equivalent without using truth table.
2.Determine whether the compound proposition ~(p∨q)∨(~p∧q)∨p to tautology.
3.Determine whether (p → q)∧(p → r)≡p → (q ∧r) using a truth table

a) We have the following set of information about a computer program, find the mistake in the program using Rules of Inferences.
i. Either a variable is not declared or there is a syntax error in the fifth line.
ii. If there is a syntax error in the fifth line, then there is a missing semicolon or there is a mistake in variable name.
iii. There is not a missing semicolon.
iv. There is a mistake in variable name.

Suppose that the domain of the propositional function P(x) consists of the integers 1, 2, 3, 4, and 5. Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions.
a) ∃xP(x)
b) ∀xP(x)
c) ¬∃xP(x)
d) ¬∀xP(x)
e) ∀x((x=3) → P(x))∨∃x¬P(x)

b) What is the value of x after each of these statements is encountered in a computer program, if x =3 before the statement is reached?
i) if x +2=5 then x := 3*x +5
ii) if (x +1=4) OR (2x +2=3) then x := x +1
iii) if (2x +3=5) AND (3x +4=13) then x := x^2 +x+7
iv) if (x +1=4) XOR (x +2=5) then x := x +1
v) if x<2 then x := 4*x^3

a) Let p, q, and r be the propositions
p : You study hard.
q : You are on merit
r : You do not get degree from Iqra University.
Express each of these propositions as an English sentence.
i) ¬q ↔ r ii) q →¬ r iii) p∨q ∨¬r iv) (p →¬r)∨(q →¬r) v) (p∧q)∨(¬q ∧¬r)

How many numbers are divisible by 2, 5, 9, and 13 between 100 and 100,000? How to implement this question in discrete math ?

Prove that log2 3 is irrational.

There are two webcams, X and Y that can automatically switch On or Off at any given time to record
a live video streaming session. To ensure the smooth recording session, the live feed input system
is programmed with certain criteria. The conditions set are; if webcam X and webcam Y are switch
on, the input system can enable the recording input and the live session will start. If either webcam
X or webcam Y is switched on, the input system cannot enable the recording and the live session
will stop. In all other conditions, the input system will enable the recording and the live session
streaming will commence.
c. Translate the pseudocodes in part (a.) and (b.) into propositional logic statements.

Show graphically that which of the following is one to one function
(1)f(x)=ln(x)
(2)g(x)=e^x
(3) h(x)=x³