Lecture 8 - Exercises
Part A - Modular Congruence
True or false:
- \(9 \equiv 1 \pmod{4}\)
- \(10 \equiv 20 \pmod{2}\)
- \(20 \equiv 4 \pmod{8}\)
- \(22 \equiv 8 \pmod{6}\)
- \(\forall x \in \mathbb{Z}, n \in
\mathbb{N}, x \equiv x \mod n\)
- \(\forall n \in \mathbb{N}, 3n \equiv 0
\pmod{n}\)
Part B - Contrapositive
Proof
Prove the following proposition using contrapositive proof.
For \(x \in \mathbb{R}\), if \(x^2 + 5x < 0\), then \(x < 0\).
Try to prove the proposition from #1 using direct proof; if you
reach a point where you’re convinced that this approach is harder, you
may give up.
Prove that if \(a\) and \(b\) have the same remainder when divided by
\(n\), then \(a \equiv b \pmod{n}\).
Part C - More Contrapositive
Proof
Proposition: Suppose \(a,
b \in \mathbb{Z}\). If both \(ab\) and \(a+b\) are even, then both \(a\) and \(b\) are even.
- Spend a few moments trying to prove this with direct proof. If you
get stuck, stop and move on to #2.
- Prove the above proposition using contrapositive proof.
Part D - If you have time
- Prove that any odd number is the difference of two squares. (For
example, \(7 = 4^2 - 3^2\).)
Book of Proof’s Style Guide
- Begin each sentence with a word, not a mathematical symbol.
- End each sentence with a period, even when the sentence ends with a
mathematical symbol or expression.
- Separate mathematical symbols and expressions with words.
- Avoid misuse of symbols.
- Avoid using unnecessary symbols.
- Use the first person plural.
- Use the active voice.
- Explain each new symbol.
- Watch out for “it”, unless it’s completely unambiguous what “it”
refers to.
- Since, because, as, for, so are used to mean that P is true (or
assumed to be true) and as a consequence Q is true also.
- Thus, hence, therefore, consequently precede a statement that
follows logically from previous sentences or clauses.
- Clarity is the gold standard of mathematical writing.
Definitions
Definition: An integer \(n\) is even if \(n = 2a\) for some integer \(a \in \mathbb{Z}\).
Definition: An integer \(n\) is odd if \(n
= 2a + 1\) for some integer \(a \in
\mathbb{Z}\).
Definition: Suppose \(a\) and \(b\) are integers. We say that \(a\) divides \(b\), written \(a
| b\), if \(b = ac\) for some
\(c \in \mathbb{Z}\). In this case we
also say that \(a\) is a
divisor of \(b\), and
that \(b\) is a
multiple of \(a\).
Definition: Two integers have the same
parity if they are both even or they are both odd. Otherwise
they have opposite parity.
Definition: Given integers \(a\) and \(b\) and \(n \in
\mathbb{Z}\), we say that \(a\)
and \(b\) are congruent
modulo \(\mathbf{n}\) if \(n \mid (a - b)\). We write this as \(a \equiv b \pmod{n}\). If \(a\) and \(b\) are not congruent modulo \(n\), then we write \(a \not\equiv b \pmod{n}\).