Set Theory - Algebra-1

1. Fundamental Concepts

Definition: A set is a well-defined collection of distinct objects, called elements or members. Sets are typically denoted by capital letters (e.g., A, B, C), and elements are listed inside curly braces { }. e.g., A = {a, b, c}.
Sets can include numbers, letters, or even other sets. Order and repetition of elements do not matter: {1, 2, 3} is the same as {3, 2, 1} or {1, 2, 2, 3}.

Union ( $$\cup$$ ): A $$\cup$$ B includes all elements in A or B (or both).

Intersection ( $$\cap$$ ): A $$\cap$$ B includes only elements common to both A and B.

Special Sets:

Empty set (∅ or { }): A set with no elements.

Universal set (U): Contains all elements under discussion.

Complement (A' or Aᶜ): The complement of set A (relative to the universal set U) contains all elements in U that are not in A.

Notation: x ∈ A means "x is an element of set A." x ∉ A means "x is not an element of set A."

Let A = {1, 2, 3} and B = {3, 4, 5} within the universal set U = {1, 2, 3, 4, 5, 6}. Find: A $$\cup$$ B, A $$\cap$$ B, and A'.

Find A $$\cup$$ B:
A $$\cup$$ B includes all elements that are in A, in B, or in both.
Their union combines all unique elements. So, A $$\cup$$ B: = {1, 2, 3, 4, 5}
Find A $$\cap$$ B:
A $$\cap$$ B includes only elements that are in both A and B.
The only common element is 3. So, A $$\cap$$ B = {3}
Find A' (complement of A):
A' contains all elements in the universal set U that are not in A.
Remove all elements of B from U. So, A' (complement of A) = {4, 5, 6}

Problem: Represent the set of all even natural numbers less than 10 using set-builder notation.

Identify the elements: 2, 4, 6, 8
Use set-builder notation: { $${x\in \mathbb{N} |x\text{ is even and }x\lt 10}$$ }

Visual Strategy: Use Venn diagrams to visualize the relationships between different sets.
Avoid Duplicates: When listing elements, each item in a set is unique.
Complement Clarity: Always confirm the universal set U before finding A'.
Empty Set Caution: ∅ is a subset of every set, but it’s not an element unless explicitly listed (e.g., {∅}).