1. Prove or disprove the following conjectures.

(a)There is an even integer which can be expressed as the sum of three even integers.

(b)Every even integer can be expressed as the sum of three even integers.

(c)There is an odd integer which can be expressed as the sum of two odd integers.

(d)Every odd integer can be expressed as the sum of three odd integers.

2.To get a feel about whether a claim is true or false, try out some examples. If you believe a claim is false, provide a specific counterexample. If you believe a claim is true, give a (formal) proof.

In this question, you should use the following definition of the rational numbers.

Definition. A real number x is rational if it may be written in the form x = p where p is an

integer and q is a positive integer. x is irrational if it is not rational. Prove or disprove the following conjectures.

Conjecture (1). If x and y are real numbers such that 3x + 5y is irrational, then at least one of x and y is irrational.

Conjecture (2). If x and y are rational numbers, then 3x + 4xy + 2y is rational. Conjecture (3). If x and y are irrational numbers, then 3x + 4xy + 2y is irrational.