A very large part of mathematics is being able to prove statements. But what are proofs? A mathematical proof is a rigorous argument made to convince other people of the absolute truth of a statement.

A big part of being able to understand and prove statements is understanding logic at its core. Unfortunately, the English language can be vague and often, words can have multiple meanings, leading to ambiguity. Thus, we need to give very precise meanings to the words we use when presenting a proof.

In mathematics, we study statements. They are either true, or false, but never both. For example, the statement "6 is an even integer" is true, and the statement "4 is an odd integer" is false. Often, '\(p\) and '\(q\)' are used to denote statements.

To combine and modify statements, we use what are called logical operators. The basic ones are NOT, AND, OR and IF... THEN. Here, we will give precise definitions of these.

**NOT**: If \(p\) is a statement, then the statement 'not \(p\)' is defined to be

> true, when \(p\) is false;

> false, when \(p\) is true.

**AND**: If \(p\) and \(q\) are statements, then the statement '\(p\) and \(q\)' is defined to be

> true, when \(p\) and \(q\) are

*both* true;

> false, when \(p\) is false, or \(q\) is false, or both \(p\) and \(q\) are false.

**OR**: If \(p\) and \(q\) are statements, then the statement '\(p\) or \(q\)' is defined to be

> true, when \(p\) is true, or \(q\) is true, or \(p\) and \(q\) are both true (or in other words, at least one of \(p\) and \(q\) are true);

> false, when \(p\) and \(q\) are both false.

*Note: in English, we sometimes use 'or' in the sense that the statement '\(p\) or \(q\)' is true when either \(p\) is true, or \(q\) is true, but not both. In mathematics, this will never be the case. In fact, we give a new word to that modifier, but we won't get into that here.***IF... THEN**: If \(p\) and \(q\) are statements, then the statement 'If \(p\) then \(q\)' is defined to be

> true, when \(p\) and \(q\) are both true, or \(p\) is false;

> false, when \(p\) is true and \(q\) is false.

*Note: the 'If... then' statement can be quite confusing at first, especially in trying to understand why the 'If... then' is always true whenever \(p\) is false. However, when \(p\) is false, we say that the 'If... then' statement is vacuously true. (Feel free to ask me anything about that).*Okay, now we just need a few more tools, and then you should be ready to do the questions.

Let's develop some number and set theory. A

*set* is a collection of objects. The objects in a set are called

*elements*. Sets are always denoted by curly brackets. For example, \(\{2,\ 4,\ 6, 8\}\) is a set. It contains the elements 2, 4, 6 and 8. To say that an object is contained in a set, we use the element symbol: \(\in\). For example, \(6\in \{2,\ 4,\ 6, 8\}\), but \(5\notin \{2,\ 4,\ 6, 8\}\).

There are some very specific number sets we would like to make use of. You have probably heard of these before, but not seen definitions of these.

**Natural Numbers**: \(\mathbb{N}=\{1,\ 2,\ ...\}\) These are the counting numbers starting from 1.

**Integers**: \(\mathbb{Z}=\{...\ -1,\ 0,\ 1,\ ...\}\) These include the natural numbers, their negatives and 0.

**Rational Numbers**: \(\mathbb{Q}=\left\{a/b\ |\ a,b\in\mathbb{Z}\ \text{and}\ b\neq 0\right\}\) These are all the possible numbers \(a/b\) such that \(a\) and \(b\) are integers and \(b\) is not 0.

**Real Numbers**: \(\mathbb{R}\) (its construction is too difficult to develop here) These are all numbers on the real number line, including ones that are irrational like \(\pi\) and \(\sqrt{2}\).

This should be enough. I've ordered the questions in such a way so that you are led to the answer