Problem 1) Let the following predicates be given:

. . .B(x) = "x is boring"

. . .H(x) = "x is a historical essay"

. . .E(x) = "x is expensive"

. . .M(x, y) = "x is more interesting than y"

(a) Write the following statements in predicate logic:

. . .1. All historical essays are boring.

. . .2. There are some boring books that are not historical essays.

. . .3. All boring historical essays are expensive.

(b) Write the following predicate logic statement in everyday English. Don't just give a word-for-word translation; your sentence should make sense.

. . .

(c) Formally negate the statement from part (b). Simplify your negated statement so that no quantifier lies within the scope of a negation.

I have finished Problem 1 part a, but the rest is a blur. I am really lost and any help will be great. Thanks. I solved Problem 1, but i have no idea how to solve problem 2,3,4.

Problem 2) In this problem, all variables range over the set of all integers. Recall that the relation a | b (read "a divides b") is defined as

1. Formulate the following statement in English and prove that it is true (for all x, a, b):

. . .

Using common precedence rules, the above statement should be interpreted as

. . .

2. Same as (1), but for the statement

. . .

3. Prove again the statement in (2), but rather than proving it from scratch, give a proof using the statement in part (1).

Problem 3) For each of the following statements:

. . .Formulate the statement in prepositional logic. You may use all of the

. . .standard logic connectives and quantifiers. You may also use integer

. . .constants (1, 2, 3, etc.), arithmetic operations (+, ×, etc.), the

. . .predicate , and the relation a|b.

. . .State if the statement is true or false.

. . .Prove your answer correct; i.e., prove either the statement or the

. . .negation of the statement. In all statements, the variables range of

. . .the set of nonnegative integers.

1. The product of any two odd integers is odd.

2. For any two numbers x and y, the sum x + y is odd if and only if the product x × y is odd.

3. For all integers a, b, and c, if a divides b and b divides c, then a divides c.

Problem 4) Also in this problem, all variables range over the nonnegative integers. Recall the definition of "prime":

. . .

i.e., a number bigger than 1 is prime if and only if it is only divisible by 1 and itself. A number is composite if it can be written as the product of smaller numbers:

. . .

Prove that any number bigger than 1 is prime if and only if it is not composite. You can break up your proof into two parts as follows:

. . .1. First prove that for any n > 1, if n is prime, then n is not composite.

. . .2. Next prove that for any n > 1, if n is not composite, then n is prime.

In the solution to this problem you can use (without proof) the fact that for any positive integers x and y, if x | y, then x < y. Remember that your solution will be graded both for correctness and clarity. This is especially important for this problem, as the proof involved is a bit longer than the proofs in the previous problems.