What are the best descriptions for the set of natural numbers N and the equation ax o b n?

The question asks for the best description of the set of natural numbers $\mathbb{N}$.

• Option A: $\mathbb{N} = \{1, 2, 3, 4, ...\}$ represents the set of positive integers. This is a widely accepted definition of natural numbers, especially in number theory.
• Option B: $\mathbb{N} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$ represents the set of integers, not natural numbers.
• Option C: $\mathbb{N} = \{0, 1, 2, 3, ..., 9\}$ is a finite set, not the infinite set of natural numbers.
• Option D: $\mathbb{N} = \{0, 1, 2, 3, ..., 100\}$ is also a finite set. While some definitions of natural numbers include 0, option A is the most common definition when 0 is not explicitly included.

The best description is A.

$$\boxed{\text{A}}$$

The question asks why the equation $ax \equiv b \pmod n$ is a linear congruence.

• A congruence is "linear" if the variable $x$ has an exponent of 1. This means it is of the first degree in $x$.
• Option A: "It is of the first degree in $x$" directly explains why it is called linear.
• Option B: "$a, b,$ and $n$ are all integers" is true for any congruence, not specific to linear ones.
• Option C: "It involves $x$ and $n$" is also true for any congruence.
• Option D: "It has a unique solution" is not always true for linear congruences; the existence and uniqueness of solutions depend on $\gcd(a, n)$.

The reason it is a linear congruence is that it is of the first degree in $x$.

$$\boxed{\text{A}}$$

Given $a \equiv 3 \pmod 7$ and $b \equiv 2 \pmod 7$, we need to find $ab \pmod 7$.
To multiply in modular arithmetic, we multiply the remainders:

$$ab \equiv (3)(2) \pmod 7$$ $$ab \equiv 6 \pmod 7$$

The value of $ab$ modulo 7 is 6.

The correct option is C.

$$\boxed{\text{C}}$$

The question asks which statements are true about the Peano axioms.

• Statement I: 0 is a natural number. The most common modern formulation of the Peano axioms defines the set of natural numbers $\mathbb{N}$ starting with 0. So, under this formulation, 0 is a natural number.
• Statement II: Every natural number has a unique successor. The Peano axioms state that every natural number $n$ has a successor $S(n)$, and if $S(n) = S(m)$, then $n = m$. This implies that each natural number has one and only one successor. This statement is true.
• Statement III: Every natural number has an additive inverse in $\mathbb{N}$. An additive inverse for a number $n$ is a number $m$ such that $n+m=0$. For example, the additive inverse of 1 is -1. However, negative numbers are not part of the set of natural numbers $\mathbb{N}$. Therefore, this statement is false.

Based on the common formulation of Peano axioms, statements I and II are true, while statement III is false.
Thus, "I and II only" is the correct choice.

The correct option is D.

$$\boxed{\text{D}}$$