Question

) Consider these statements, "All hummingbirds are richly colored." "No large birds live on honey. "Birds that do not live on honey are dull in color." "Hummingbirds are small. Where, P(x): "x is a hummingbird" Q(x): "x is large." R(x): "x lives on honey." S(x): "x is richly colored." Answer the following questions, i. Express the statements logically. Using rules of inference logically prove that the last statement is a valid conclusion when the first three are premises. b) Given g(x) = (x - 3 find g(x),x > 0. c) Show that the sequence {an} is a solution of the recurrence relation an = An-1 +2. An-2 +2n - 9 if, an = 3.(-1)"+2" - n + 2 an = 7.2" - n +2 d) Find the solution to the recurrence relation an = 2.7. An-1 and initial condition a, = 1.

Answer 1

The statements can be logically expressed using universal quantifiers and implications. A step-by-step logical proof demonstrates the validity of the conclusion using rules of inference. Part B involves evaluating a function, and Part D solves a geometric recurrence relation.

Step-by-step explanation:

To express the given statements logically:

• №1: ∀x(P(x) → S(x)) - 'All hummingbirds are richly colored.'
• №2: ∀x(Q(x) → ¬R(x)) - 'No large birds live on honey.'
• №3: ∀x(¬R(x) → ¬S(x)) - 'Birds that do not live on honey are dull in color.'
• №4: ∀x(P(x) → ¬Q(x)) - 'Hummingbirds are small.'

Using rules of inference, we can show that the last statement is a valid conclusion:

1. From №2 and №3, we apply hypothetical syllogism to get: ∀x(Q(x) → ¬S(x)).
2. From №4, we know that if x is a hummingbird, then x is not large, hence if P(x) then ¬Q(x).
3. Then by modus ponens, from №1, if P(x) then S(x) (hummingbirds are richly colored).
4. Therefore, combining the results, we deduce that hummingbirds being small (¬Q(x)) and also richly colored (S(x)) is a valid conclusion given the premises.

For part b), the function is g(x) = x - 3. If x > 0, then g(x) > -3 since we subtract 3 from a positive number.

For part d), the solution to the recurrence relation an = 2.7an-1 with initial condition a1 = 1 is given by a geometric sequence where each term is 2.7 times the previous term.