Sets PDF

# Sets - **Definition:** A collection of things. - **Elements:** The individual things in a set. ## Notation - **Braces:** `{1, 2, 3...}` or `{Hello, Water, Jack}`. ## Representing the existence of an element in a set - **Membership:** `x ∈ A` means that *x* is in the set *A*. - **Non-membe...

# Sets - **Definition:** A collection of things. - **Elements:** The individual things in a set. ## Notation - **Braces:** `{1, 2, 3...}` or `{Hello, Water, Jack}`. ## Representing the existence of an element in a set - **Membership:** `x ∈ A` means that *x* is in the set *A*. - **Non-membership:** `x ∉ A` means that *x* is not in the set *A*. ## Number Sets - **Natural Numbers (N): `{1, 2, 3, 4...}`** - **Integers (Z): `{..., -2, -1, 0, 1, 2...}`** - **Rational Numbers (Q):** A number that can be expressed as a division of two integers, $a/b$ where $b \neq 0$. Includes natural numbers, integers, and any number that can be expressed as a fraction. - **Irrational Numbers:** Numbers that cannot be expressed as a division of two integers. For example: √2, π, e. - **Real Numbers (R):** All rational and irrational numbers. - **Imaginary Numbers (I):** Numbers whose square is a negative number. For example: $i^2 = -1$. - **Complex Numbers (C):** The sum of a real number and an imaginary number. For example: `2 + 3i`. ## Summarizing - These sets are nested within each other. - **Natural numbers** are always within **integers**, **integers** are always within **rational numbers,** and so on. - **Complex numbers** are the most inclusive set, containing the rest. ## Equality - **Axiom:** Two sets are equal if and only if they have the same elements. - **Order:** The order of the elements does not matter. - **Repetitions:** Repetitions are ignored. ## Set-Builder Notation - Describes a set by defining a property that each element must have. ## Types of Sets - **Universal Set (U):** Contains all the other sets. - **Empty Set (∅, { }):** The set with no elements. - **Singleton:** A set with only one element, such as `{7}` or `{1, 2, 4, 8}`. - **Finite Set:** A set with a limited number of elements. - **Infinite Set:** A set with an unlimited number of elements represented by "...". ## Subset - **Definition:** A set *B* is said to be a subset of set *A* if all elements of *B* are also elements of *A*. We write this as `B ⊆ A`. - **Proper Subset:** A set *B* is a proper subset of *A* if (a) *A* and *B* are not equal and (b) *B* is a subset of *A*. We write this as `B ⊂ A`. ## Cardinality - The number of elements in a set. Denoted by `n(A)`. - For example, the cardinality of set `{1, 2, 3, 4, 5}` is 5. ## Equivalent Sets - **Definition:**Two sets are equivalent if they have the same number of elements, even if they contain different elements. - **Notation:** `A ~ B` - **Example:** `A = {1, 2, 3}` and `B = {4, 5, 6}` are equivalent because they both have three elements. ## More About Subsets - If `A ⊂ B`, then `B` must have at least one more element than `A`. ## Power Set - **Definition:** The power set of a set A is the set of all subsets of A. It is denoted by `P(A)`. - **Example:** If `A = {1, 2}`, then `P(A) = { {}, {1}, {2}, {1,2} }`. - **Calculation:** The power set of a set with *n* elements has `2^n` elements. ## Ordered Pairs - **Definition:** A pair of elements where order matters, written as `(a, b)`. - **Example:** `(1,2)` is different from `(2,1)`. - **n-Tuples:** Ordered *n*-tuples are sequences of *n* elements, written as `(x₁, x₂, x₃,..., xₙ)`. ## The Cartesian Product - **Definition:** The Cartesian product of two sets *A* and *B* is a set of all possible ordered pairs where the first element of each pair is from *A* and the second element is from *B*. It is written as `A × B`. - **Example:** If `A = {1, 2}` and `B = {c, d}`, then `A × B = {(1, c), (1, d), (2, c), (2, d)}`. ## Properties of the Cartesian Product - **Non-commutative:** `A × B ≠ B × A` - **Associative:** `(A × B) × C = A × (B × C)` - **Distributive:** `A × (B ∪ C) = (A × B) ∪ (A × C)`. - **Example**: `{1, 2} × ({c, d} ∪ { e, f}) = ({1, 2} × {c, d}) ∪ ({1, 2} × {e, f})` ## Cartesian Plane - A two-dimensional plane where points are defined by pairs of real numbers. - The Cartesian plane has two axes: the x-axis and the y-axis. - Each point on the plane is represented by an ordered pair `(x, y)`, where *x* is the x-coordinate and *y* is the y-coordinate. ## Venn Diagrams - A way to visualize the relationship and operations between sets. - Each set is represented by a circle, and the area of overlap between circles represents the intersection of the sets. ## Relationship of Sets - **Intersection:** The intersection of two sets *A* and *B* is the set of elements that are common to both *A* and *B*. It is represented by `A ∩ B`. - **Union:** The union of two sets *A* and *B* is the set of all elements that are in at least one of *A* or *B*. It is represented by `A ∪ B`. - **Difference:** The difference of set *A* and set *B* is the set of all elements in *A* that are not in *B*. It is represented by `A \ B`. - **Complement:** The complement of a set *A* with respect to a universal set *U* is the set of all elements in *U* that are not in *A*. It is represented by `A'`. ## Set Operations - **Operations** that produce new sets. ### Properties of Union and Intersection - **Commutative:** - `A ∩ B = B ∩ A` - `A ∪ B = B ∪ A` - **Associative:** - `(A ∪ B) ∪ C = A ∪ (B ∪ C)` - `(A ∩ B) ∩ C = A ∩ (B ∩ C)` - **Distributive:** - `A ∪ (B ∩ C )= (A ∪ B) ∩ (A ∪ C)` - `A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)` - **Empty Set:** - `A ∪ { } = A` - `A ∩ { } = { }` - **Universal Set:** - `A ∪ U = U` - **Idempotent:** - `A ∪ A = A` - `A ∩ A = A` ## Difference of Sets - **Definition:** The difference of set *A* and set *B* is the set of all elements in *A* that are not in *B*. It is represented by *A − B*. - **Elements:** The elements in the difference set are all elements in `A` but not in `B`. - **Non-commutative:** `A − B ≠ B − A` - **Example:** If `A = {1, 2, 3}` and `B = {4, 5, 6}`, then `A − B = {1, 2, 3}` and `B − A = {4, 5, 6}`. ## Complement of a Set - **Definition:** The complement of a set *A* with respect to a universal set *U* is the set of all elements in *U* that are not in *A*. It is represented by `A'`. - **Example:** If `U = {1, 2, 3, 4, 5, 6, 7}` and `A = {1, 2}`, then `A' = {3, 4, 5, 6, 7}`. ## Partition of Sets - **Disjoint Sets:** Two sets are disjoint if they have no elements in common. This is denoted by `A ∩ B = ∅`. - **Partition of a Set:** A finite or infinite collection of nonempty sets that satisfies the following conditions: 1. **The union of all the sets in the partition is equal to the original set.** 2. **All of the sets in the partition are mutually disjoint.**.