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Determine the minimum number of separate racks needed to store the chemicals given in the table
(1st column) by considering their incompatibility u
sing graph coloring technique. Clearly state you
steps and graphs used.
Chemical: Incompatible with
Ammonia (anhydrous): Mercury, chlorine, calcium hypochlorite, iodine, bromine,
hydrofluoric acid (anhydrous)
Chlorine: Ammonia, acetylene, butadiene, butane, methane, propane,
hydrogen, sodium carbide, benzene, finely divided metals,
turpentine
Iodine: Acetylene, ammonia (aqueous or anhydrous), hydrogen
Silver: Acetylene, oxalic acid, tartaric acid, ammonium compounds,
pulmonic acid
Iodine: Acetylene, ammonia (aqueous or anhydrous), hydrogen
Mercury: Acetylene, pulmonic acid, ammonia
Fluorine: All other chemicals
Let x and y be real numbers. Prove that, if 5x+y>11, then x>2 or y>1.
4. (2 points) Solve these recurrence relations together with the initial conditions given. Each is worth 1 point.
a) an = 6an-1 - 11an-3 +6an-3 for n > 3, a0 = 4, a1 = 6, a2 = 12
b) an = an–1 +9an-2 – 9an-3 for n > 3, a0 = 6, a1= 0, a2 = 30
Construct a truth table for each of these compound propositions. [6 marks]
a) p ⊕ p b) p ⊕￢p
c) p ⊕￢q d) ￢p ⊕￢q
e) (p ⊕ q) ∨ (p ⊕￢q) f ) (p ⊕ q) ∧ (p ⊕￢q)
Suppose that the domain of the predicateP(x) consists of −2, −1, 0, 1, and2. Write out each
of the following predicate logic formulas in propositional logic formulas using disjunctions, conjunctions,
negations, or their combinations.
use the Euclidean algorithm to express gcd(94, 159) as a linear combination of 94 and 159
Let P(x,y) denote the sentence x2 + 1≥ x + 1. What are the truth value of the following where the domain of x and y is the set of all integers?
a. ⱯxⱯyP(x,y)
b. ⱯxƎyP(x,y)
c. ƎxⱯyP(x,y) .
d. ƎxƎyP(x,y)
3. Which of the intervals (0, 5), (0, 5], [0, 5), [0, 51, (1,4], [2, 3], (2, 3) contains a) 0? b)1?c)2? d) 3? e)4? f)5?
Solve the linear congruence 34x ≡ 53(mod 89).
Suppose that a password for a computer system must have at least 6, but no more than 9 characters, where each character in the password is a lowercase English letter, or an uppercase English letter, or a digit, or one of the five special characters *, <, >, !, and #.
(a) How many different passwords are available for this computer system?
(b) How many of these passwords contain at least one occurrence of at least one of the five special characters?
(c) Using your answer to part (b), determine how long it takes a hacker to try every possible password, assuming that it takes one microsecond for a hacker to check each possible password.
