1.2 Subsets
2 min read•june 18, 2024
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 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
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- Set theory is the mathematical study of collections of objects
- uses curly braces to enclose elements: {1, 2, 3}
- is denoted by (element of) or ∉ (not an element of)
- occurs when two sets have exactly the same elements
- include union, intersection, and complement
Subsets and proper subsets
- A 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
- contains some, but not all, elements of the original set
- Denoted as (even numbers and natural numbers)
- has no elements, denoted as 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 elements is
- Includes the empty set and the set itself
- A set with 3 elements, {a, b, c}, has subsets
- Subsets: , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}
- A set with 4 elements, {w, x, y, z}, has subsets
- Follows the pattern of doubling the number of subsets for each additional element
Set equivalence by cardinality
- is the number of elements in a set
- Denoted as for set A (size or magnitude)
- Two sets are equivalent if they have the same
- If , 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)
Key Terms to Review (20)
∈: The symbol ∈ denotes the relationship of membership in set theory, indicating that an element is a member of a set. This concept is foundational to understanding how sets are constructed, what they contain, and how elements relate to one another within the framework of set theory. Recognizing this membership relation is crucial for grasping other concepts such as subsets, operations on sets, and visual representations like Venn diagrams.
⊂: The symbol '⊂' represents a subset relationship between two sets, indicating that all elements of the first set are also elements of the second set. This concept is fundamental in understanding how sets interact and relate to one another, particularly in terms of inclusion. The subset relationship helps clarify properties of sets and allows for operations such as unions and intersections to be better understood.
⊆: The symbol '⊆' represents the concept of subsets in set theory, indicating that all elements of one set are also contained within another set. This relationship is fundamental to understanding how sets interact with each other, and it can also imply equality when both sets contain exactly the same elements. Recognizing subsets helps in various mathematical contexts, including operations involving unions and intersections, as well as in visualizing relationships between sets using diagrams.
Cardinality: Cardinality is the measure of the "number of elements" in a set. It helps compare the size of sets, even if they are infinite.
Cardinality: Cardinality refers to the number of elements in a set, which helps to determine the size of that set. It's a crucial concept because it allows us to compare different sets, understand relationships between them, and perform operations that involve those sets. Cardinality can apply to finite sets, where the number of elements is countable, as well as infinite sets, which require a different approach for comparison.
Empty set: The empty set is a fundamental concept in set theory, representing a set that contains no elements at all. It is denoted by the symbol $$\emptyset$$ or by a pair of braces, {}. This concept is crucial because it serves as a building block for more complex sets and helps to clarify the structure of subsets and relationships between sets.
Power set: A power set is the set of all possible subsets of a given set, including the empty set and the set itself. It reflects the idea that any set can be broken down into different combinations of its elements, highlighting the concept of subsets and their relationships within set theory.
Proper subset: A proper subset is a set that contains some, but not all, elements of another set. This means that if set A is a proper subset of set B, then every element of A is also an element of B, but B must have at least one element that is not in A. Understanding proper subsets helps in grasping the relationships between sets and the concept of subset hierarchies.
Sequences: Sequences are ordered lists of elements where the position of each element is significant. They can be finite or infinite and often follow a specific pattern or rule.
Series: A series is the sum of the terms of a sequence. In math, it often involves summing an infinite number of terms based on a specific rule or formula.
Set equality: Set equality occurs when two sets contain exactly the same elements, regardless of the order or duplication of those elements. This concept emphasizes that two sets are considered equal if they have identical members, which plays a crucial role in understanding relationships between sets and their properties.
Set membership: Set membership refers to the relationship between an element and a set, indicating whether the element is contained within that set. This concept is fundamental in mathematics, as it allows us to classify elements and understand their relationships to various collections of objects. The notation for set membership typically uses the symbol '$$\in$$' to denote that an element belongs to a set, while '$$\notin$$' indicates that it does not.
Set notation: Set notation is a standardized way of representing sets and their elements using specific symbols and conventions. It provides a clear language for defining sets, subsets, and relationships between sets, facilitating communication and understanding in mathematics. This notation includes symbols such as brackets, curly braces, and various mathematical operations to convey the properties and membership of elements within sets.
Set operations: Set operations are mathematical procedures used to combine, compare, or manipulate sets, which are collections of distinct objects. The primary operations include union, intersection, and difference, each serving a unique purpose in understanding the relationships between sets. These operations enable the analysis of how sets relate to one another and are foundational in various mathematical contexts.
Set theory: Set theory is a branch of mathematical logic that studies sets, which are collections of objects. It provides a foundational framework for various mathematical concepts and operations, including relationships between different groups, classifications, and how elements interact within those groups. This framework is crucial for understanding concepts like subsets, Venn diagrams, and various set operations, which are fundamental in both theoretical and applied mathematics.
Subset: A subset is a set where all elements are also contained within another set. If A is a subset of B, every element of A is also an element of B.
Subset: A subset is a set that contains some or all elements of another set. This concept is fundamental in understanding relationships between sets and helps in visualizing how different sets can relate to one another. Recognizing subsets is key to performing operations involving multiple sets and understanding their relationships through diagrams and various operations.
Universal set: The universal set is the set that contains all objects or elements under consideration, typically denoted by the symbol U. It serves as the reference set in Venn diagrams and includes every element of the sets being discussed.
Universal Set: The universal set is the set that contains all the objects or elements under consideration for a particular discussion or analysis. It serves as the foundation for defining other sets, as all subsets are drawn from this larger set, helping to establish relationships and operations between different collections of items.
Venn diagrams: Venn diagrams are visual representations used to show the relationships between different sets, illustrating how they intersect, differ, or share elements. They are particularly useful in understanding concepts like subsets, where one set is wholly contained within another, and in analyzing conditional probability, where the relationship between events can be clearly depicted. By using circles to represent sets, Venn diagrams make it easier to visualize complex logical relationships.