Set Basics 1832Ac

Question: User: SET And Their OPERATIONS Definitions • Diagrams • Laws • Proofs • Applications SET THEORY What Is a Set? A set is a well-defined collection of distinct objects. Each object in the collection is called an element or member of the set. Example: V = {a, e, i, o, u} V contains the five vowels of the English alphabet. Sets may contain numbers, letters, people, ordered pairs or other sets. SET THEORY What Does “Well-Defined” Mean? A collection is well-defined when we can decide clearly whether a particular object belongs to it. Membership must not depend on personal opinion. Well-defined: A = {2, 4, 6, 8, 10}, the positive even numbers up to 10. Not well-defined: “the beautiful flowers”, without a fixed criterion. For A, 4 belongs to the set and 7 does not. The rule gives a definite answer. SET THEORY Distinct Elements and Order A set records which elements are present. Repeated entries do not create new elements, and changing the order does not change the set. {1, 2, 2, 3} = {1, 2, 3} {1, 2, 3} = {3, 1, 2} The distinct letters of APPLE form {A, P, L, E}. Cardinality means the number of distinct elements. Here, |{A, P, L, E}| = 4. SET THEORY Set Notation and Membership Sets are usually named with capital letters, such as A or B. Curly brackets contain the elements. The symbols ∈ and ∉ express membership. A = {2, 4, 6, 8} 4 ∈ A means “4 is an element of A”. 5 ∉ A means “5 is not an element of A”. Membership compares an element with a set. A subset compares one set with another. SET THEORY Roster / Tabular Form In roster or tabular form, we list the distinct elements inside curly brackets and separate them with commas. Positive factors of 12: F = {1, 2, 3, 4, 6, 12} English vowels: V = {a, e, i, o, u} Positive odd numbers: O = {1, 3, 5, 7, …} The dots (…) indicate that a clear pattern continues. SET THEORY Set-Builder Form Set-builder form describes the elements using a condition. The vertical bar | is read as “such that”. A = {x ∈ ℕ | 1 ≤ x ≤ 5} = {1, 2, 3, 4, 5} Read: all natural numbers x such that 1 ≤ x ≤ 5. E = {x ∈ ℤ | x = 2k for some k ∈ ℤ} defines the even integers. Convention in these slides: ℕ = {1, 2, 3, …}; ℤ includes negative integers and zero. SET THEORY Descriptive Form Descriptive form states in words which objects belong to the set. The description must identify the elements clearly. Descriptive: A is the set of the first five natural numbers. Roster: A = {1, 2, 3, 4, 5} Set-builder: A = {x ∈ ℕ | 1 ≤ x ≤ 5} These three forms represent exactly the same set. SET THEORY How Sets Are Classified Some types describe the size or role of one set. Other terms describe a relationship between two sets. A set may fit more than one description. Size: empty, singleton, finite or infinite. Role: universal set or power set. Relationships: equal, equivalent, subset, proper subset, disjoint or overlapping. For example, {7} is both a singleton set and a finite set. SET THEORY Empty Set — Definition An empty set contains no elements. It is denoted by ∅ or { }. Its cardinality is zero. A = {x ∈ ℕ | x < 1} = ∅ There is no positive natural number less than 1. ∅ means no elements; it is not the same as {0}. SET THEORY Empty Set —Examples An empty set contains no elements. It is denoted by ∅ or { }. Its cardinality is zero. B = {x ∈ ℤ | x² = −1} = ∅ No integer has a square equal to −1. {∅} contains one element: the empty set itself. |∅| = 0, but |{∅}| = 1 and |{0}| = 1. SET THEORY Singleton Set — Definition A singleton set contains exactly one distinct element. Its cardinality is one. A = {7}, so |A| = 1. B = {x ∈ ℤ | x + 3 = 5} = {2}. A singleton set is always finite. SET THEORY Singleton Set —Examples A singleton set contains exactly one distinct element. Its cardinality is one. C = {0} is a singleton set. D = {a, a, a} = {a}, so D is also a singleton. E = {∅} contains one element and is a singleton. Count distinct members, rather than repeated entries. SET THEORY Finite Set — Definition A set is finite if it has a whole-number quantity of elements. We can complete a count of all its distinct members. A = {2, 4, 6, 8, 10}, so |A| = 5. V = {a, e, i, o, u}, so |V| = 5. The empty set is finite, with zero elements. SET THEORY Finite Set Examples A set is finite if it has a whole-number quantity of elements. We can complete a count of all its distinct members. F = {1, 2, 3, 4, 6, 12} has six elements. G = {1, 2, 3, …, 100} has 100 elements. A large set can still be finite if its count has an endpoint. Dots do not always mean infinity: {1, 2, …, 100} is finite. SET THEORY Infinite Set — Definition An infinite set is a set that is not finite. Its elements cannot all be counted in a process that ends after finitely many steps. ℕ = {1, 2, 3, …} E = {2, 4, 6, 8, …} Each listed number is followed by another member. There is no largest natural number and no largest positive even number. SET THEORY Infinite Set — Examples An infinite set is a set that is not finite. Its elements cannot all be counted in a process that ends after finitely many steps. ℤ = {…, −2, −1, 0, 1, 2, …} is infinite. ℝ, the set of all real numbers, is also infinite. ℕ and ℤ are countably infinite; ℝ is uncountably infinite. At university level, infinite sets can have different cardinalities. SET THEORY Universal Set — Definition A universal set U contains all elements under consideration in a particular discussion. Every set being discussed must be a subset of U. U = {1, 2, 3, 4, 5, 6} A = {2, 4, 6} and B = {1, 3, 5} Both A and B are subsets of U. The universal set depends on the context of the problem. SET THEORY Universal Set — Examples A universal set U contains all elements under consideration in a particular discussion. Every set being discussed must be a subset of U. For a class survey, U may be all students in that class. For a die roll, U = {1, 2, 3, 4, 5, 6}. If A = {2, 4, 6}, its complement in this U is {1, 3, 5}. Changing U can change the complement of A. SET THEORY Equal Sets — Definition Two sets are equal if they contain exactly the same elements. We write A = B. Every member of each set must belong to the other. A = {1, 2, 3} and B = {3, 1, 2} A = B because their elements are identical. Order and repetition do not affect equality. SET THEORY Equal Sets — Examples Two sets are equal if they contain exactly the same elements. We write A = B. Every member of each set must belong to the other. A = {a, b, c} and B = {c, b, a, a} are equal. C = {1, 2, 3} and D = {1, 2, 4} are not equal. C and D have the same size but different members. Equal sets always have equal cardinalities. SET THEORY Equivalent Sets — Definition Two sets are equivalent if their elements can be paired one-to-one with none left over. For finite sets, this means equal cardinalities. A = {a, b, c} and B = {1, 2, 3} Pair a with 1, b with 2, and c with 3. Thus |A| = |B| = 3, although A ≠ B. Equivalent sets may have different elements. SET THEORY Equivalent Sets — Examples Two sets are equivalent if their elements can be paired one-to-one with none left over. For finite sets, this means equal cardinalities. ℕ = {1, 2, 3, …} and E = {2, 4, 6, …} are equivalent. The rule n ↦ 2n pairs every natural number with one positive even number. For infinite sets, equivalence requires a bijection, not a finite count. SET THEORY Subset — Definition A is a subset of B if every element of A is also an element of B. We write A ⊆ B. A subset is allowed to equal B. A = {2, 4}, B = {1, 2, 3, 4, 5} A ⊆ B because both 2 and 4 belong to B. Formal condition: for every x, if x ∈ A then x ∈ B. SET THEORY Subset — Examples A is a subset of B if every element of A is also an element of B. We write A ⊆ B. A subset is allowed to equal B. If B = {1, 2, 3}, then {1, 3} ⊆ B and B ⊆ B. ∅ ⊆ B because the empty set has no element outside B. {1, 4} is not a subset of B because 4 ∉ B. 2 ∈ {1, 2, 3}, whereas {2} ⊆ {1, 2, 3}. SET THEORY Proper Subset — Definition A is a proper subset of B if every element of A belongs to B and A ≠ B. We write A ⊊ B. B contains at least one element outside A. A = {1, 2}, B = {1, 2, 3} A ⊊ B because 3 belongs to B but not to A. A ⊊ B means A ⊆ B and A ≠ B. SET THEORY Proper Subset — Examples A is a proper subset of B if every element of A belongs to B and A ≠ B. We write A ⊊ B. B contains at least one element outside A. If A = B = {1, 2}, then A ⊆ B but A is not a proper subset. ∅ is a proper subset of every nonempty set. For finite sets, A ⊊ B implies |A| < |B|. For infinite sets, a proper subset may have the same cardinality as its parent set. SET THEORY Disjoint Sets — Definition Two sets are disjoint if they have no common elements. Their intersection is the empty set: A ∩ B = ∅. A = {1, 3, 5}, B = {2, 4, 6} No number belongs to both A and B. Therefore A ∩ B = ∅. Disjointness describes a relationship between two sets. SET THEORY Disjoint Sets — Examples Two sets are disjoint if they have no common elements. Their intersection is the empty set: A ∩ B = ∅. Positive integers and negative integers are disjoint. {a, b} and {b, c} are not disjoint because both contain b. The empty set is disjoint from every set. For finite disjoint sets, |A ∪ B| = |A| + |B|. SET THEORY Overlapping Sets — Definition Two sets overlap when they have at least one common element. Their intersection is nonempty: A ∩ B ≠ ∅. A = {1, 2, 3, 4}, B = {3, 4, 5} A ∩ B = {3, 4} The common elements are 3 and 4. Overlapping sets may have different sizes. SET THEORY Overlapping Sets — Examples Two sets overlap when they have at least one common element. Their intersection is nonempty: A ∩ B ≠ ∅. M = {Ali, Sara, Ahmed}, C = {Sara, Hina} Sara belongs to both groups, so M ∩ C = {Sara}. Neither group is a subset of the other in this example. Some textbooks use “overlapping” more narrowly when neither set contains the other. make its easer