Lecture 10/08

Definition of Cardinality: The cardinality of a set is a measure of the "number of elements" in the set.
Types of Cardinality:
- Countable Sets: Sets that can be put into a one-to-one correspondence with the natural numbers
- Including finite sets. Examples: Natural numbers (N), Integers (Z).
- Uncountable Sets: Sets that cannot be put into one-to-one correspondence with the natural numbers. Examples include the set of real numbers (R).

Countable vs. Uncountable:
- Claim: The set of real numbers (R) is not countable.
- New Notation: The cardinality of the continuum is often denoted as $c$ or sometimes referred to as "little c".
Power Set Cardinality:
- If A is a set, then the power set of A, denoted as $P(A)$ , has strictly greater cardinality than A.
- This result allows for constructing larger sets beyond the original.

If $A ext{ is a subset of } B$ : then $|A| ext{ (cardinality of A)} \leq |B| ext{ (cardinality of B)}$ .
Injection (One-to-One Function): If there exists an injection from A to B, then $|A| \leq |B|$ .
Surjection (Onto Function): If there exists a surjection from A to B, then $|A| \geq |B|$ .
For countable sets A and B, the union $A \bigcup B$ is countable.
- If A is countable and B is uncountable, then $A \bigcup B$ is also uncountable.

Bernstein's Theorem: If $|A| \leq |B|$ and $|B| \leq |A|$ , then $|A| = |B|$ .
- Although it seems intuitive, proving this for infinite sets is complex.
Cardinalities of Intervals:
- Example: The interval (0,1) has the same cardinality as (0,2), since we can define a simple bijection, such as scaling by a factor of 2.

Example: To show that $|[0, 1]| = |[0, 2]|$ by defining the function $f(x) = 2x$ .
Set Operations:
- Explore the cardinality of $Z^2$ (the set of all pairs of integers) to determine whether it is finite, countably infinite, or uncountable.
- For integers $z$ , pairs can be listed systematically to prove that $|Z^2|$ is countably infinite.

Density of Rational Numbers: Rational numbers are dense in the real number line. This means that within any interval of real numbers, no matter how small, there exist infinitely many rational numbers.
Measure of Sets: When examining measures on intervals,
- The set of rational numbers has measure 0 compared to the real numbers, implying that the irrationals substantially dominate the reals.

Definition of Divisibility:
- Notation: $a | b$ means that a divides b (b is divisible by a) if there exists an integer k such that $b = ak$ .
Properties of Divisibility:
- If $a | b$ and $a | c$ , then $a | (b + c)$ is provable through the definitions of divisibility and some algebraic manipulations.
Corollary of Divisibility:
- If $m$ and $n$ are integers, then a linear combination of multiples of these integers also retains the property of divisibility by a common factor.

Definition of Congruence:
- Two integers a and b are congruent modulo m (denoted $a \equiv b \; (mod \, m)$ ) if $m | (a - b)$ .
Equivalence Classes: Each equivalence class under a modulus contains all integers that yield the same remainder when divided by that modulus.
- Example: $25 \equiv 1 \; (mod \, 3)$ because the subtraction yields a multiple of 3.
Arithmetic Operations in Modular Arithmetic:
- The rules governing addition, subtraction, and multiplication in modular arithmetic closely mimic those of standard arithmetic, with adjustments applied through mod.

Understanding cardinality, properties of sets, and modular arithmetic is essential for delving into deeper mathematical concepts and proofs.
Gain proficiency in identifying whether sets are finite, countable, or uncountable, which helps establish foundational knowledge in discrete mathematics and number theory.