Urgent Homework

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- Translate the given English sentence into a compound proposition using logical operators and the letters
*b,c,h*as follows:

*b *: The bus is on time.

*c *: I catch the bus.

*h *: I leave home early.

- (2 points) I will catch the bus if I leave home early or the bus is late.
- (2 points) If missing the bus implies I left home late, then catching the bus implies I left home early.

- Translate the given English sentence into predicate logic using logical operators, quantifiers, and the predicates
*C,H,L*as follows:

*C*(*x*) : *x *is a cute kitten.

*H*(*x*) : *x *is a human.

*L*(*x,y*) : *x *loves *y.*

- (2 points) No human is a cute kitten.
- (2 points) Every cute kitten loves some human.

- (4 points) Calculate the following sum:

- (4 points) For
*n*∈ Z+, let the set of proper factors of*n*be defined as

*F _{n }*:= {

For example, *F*_{12 }= {1*,*2*,*3*,*4*,*6}.

**Prove or disprove the following statement: **There exists an integer *n *such that

- (4 points) You are on an island where every person is either a knight or a knave. Knights always tell the truth and knaves always lie. You encounter Alaric and Benjin. Alaric says “Benjin is a knight” and Benjin says “Either I am a knave or Alaric is a knave, but not both”. What can you conclude about the identities of Alaric and Benjin? Show your work.
- (6 points) Prove that
- Consider the function

*g *: *A *→ *B*

*x *7→ 3*x*^{2 }− 6

where *A *and *B *are subsets of R.

- (2 points) If
*A*= R, and*B*= R, is*g*injective (one-to-one)? Justify your answer. - (2 points) If
*A*= R, choose a codomain*B*⊆ R so that*g*is surjective (onto). Justify your choice of*B*. - (2 points) Let
*S*= [0*,*1]. What is*g*(*S*)?

- (6 points) Use the definition of even/odd numbers to prove that
*x*is even**if and only if**3*x*+ 2 is even.

- (6 points) Find the solution (closed form formula of
*a*) to this recurrence relation with the given initial condition:_{n}*a*=_{n }*a*_{n}_{−1 }− 2*n*,*a*_{0 }= 3.

(6 points) Assume that √21 is irrational, prove that √3 + √7 is irrational.

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