Section 1.1 Prerequisites - Set Theory and Mathematical Symbols

Definition and Types of Sets

A set is defined as a well-defined collection of objects with no duplication. The individual items within a set are referred to as elements or members. Sets are broadly categorized based on the quantity of elements they contain into finite sets and infinite sets.

A finite set contains a fixed, countable number of distinct elements. For example, the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ represents a finite set containing the single-digit non-negative integers.

An infinite set contains an unending collection of elements that continues indefinitely. For example, the set $D = \{1, 2, 3, \dots\}$ represents an infinite set containing all positive integers.

Symbol List and Number Systems

Mathematical notation relies on specific standardized symbols to represent standard sets of numbers, logical operations, and set relations.

The symbol $\emptyset$ denotes the empty set, which is the unique set containing no elements. This can also be expressed as $E = \emptyset$.

The symbol $N$ represents the set of natural numbers.

The symbol $Z$ represents the set of integers, which includes all positive whole numbers, negative whole numbers, and zero.

The symbol $Q$ represents the set of all rational numbers, which consist of numbers that can be written as a ratio of two integers.

The symbol $Q^c$ represents the set of all irrational numbers, which consist of numbers that cannot be expressed as a ratio of two integers and have non-repeating, non-terminating decimal expansions.

The symbol $R$ represents the set of real numbers, encompassing all rational and irrational numbers.

The symbol $C$ represents the set of complex numbers.

The symbol $\in$ indicates membership in a set, meaning is an element of or belongs to, as demonstrated in the statement $a \in A$.

The symbol $\exists$ is the existential quantifier, meaning there exist or there exists.

The symbol $\forall$ is the universal quantifier, meaning for all or for every.

The vertical bar symbol $\mid$ means such that.

Homework Problem Solutions and Analysis

Problem $13$ determines the nature of a decimal expansion, concluding that the decimal terminates (Yes, it terminates).

Problem $14$ addresses fractional expressibility and perfect squares. A number can be written as a simple fraction, and the decimal value $0.16$ is specifically identified as a perfect square of a simple fraction, since $\sqrt{0.16} = 0.4 = \frac{2}{5}$.

Problem $15$ classifies a number within the hierarchy of number sets, identifying it as belonging simultaneously to the set of integers, the set of rational numbers, and the set of real numbers.