1.2 Subsets

Set theory is all about organizing things into groups. It's like sorting your stuff into boxes, where each box is a set. We use special symbols to show what's in each set and how they relate to each other.

Subsets are like mini-groups within bigger groups. For example, apples are a subset of fruits. We can figure out how many subsets a group has and compare different groups to see if they're the same size.

Set Theory and Subsets

Set Theory Fundamentals

• Set theory is the mathematical study of collections of objects
• Set notation uses curly braces to enclose elements: {1, 2, 3}
• Set membership is denoted by ∈ (element of) or ∉ (not an element of)
• Set equality occurs when two sets have exactly the same elements
• Set operations include union, intersection, and complement

Subsets and proper subsets

• A subset contains some or all elements of another set
  • If set A is a subset of set B, every element in A is also in B (fruits and apples)
  • Denoted as $A \subseteq B$
• Proper subset contains some, but not all, elements of the original set
  • Denoted as $A \subset B$ (even numbers and natural numbers)
• Empty set has no elements, denoted as $\emptyset$ or {}
  • The empty set is a subset of every set, including itself (universal subset)

Calculating total subsets

• The number of subsets for a set with $n$ elements is $2^n$
  • Includes the empty set and the set itself
• A set with 3 elements, {a, b, c}, has $2^3 = 8$ subsets
  • Subsets: $\emptyset$, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}
• A set with 4 elements, {w, x, y, z}, has $2^4 = 16$ subsets
  • Follows the pattern of doubling the number of subsets for each additional element

Set equivalence by cardinality

• Cardinality is the number of elements in a set
  • Denoted as $|A|$ for set A (size or magnitude)
• Two sets are equivalent if they have the same cardinality
  • If $|A| = |B|$, then A and B are equivalent (same number of elements)
• Equivalent sets can have different elements but the same number of elements
  • {a, b, c} and {1, 2, 3} are equivalent sets (both have cardinality of 3)
  • {dog, cat, bird} and {red, blue, green} are equivalent (same size, different elements)