Sets. The Building Blocks Of Mathematics. IntroductionSets are fundamental mathematical objects that represent collections of elements. They are used to describe and analyze relationships between different objects or concepts. In this article, we will explore the basics of sets, including their notation, operations, and applications.Definition of a SetA set is a collection of distinct objects or elements. These elements can be anything, from numbers and letters to geometric shapes or even other sets. Sets are usually denoted by curly braces {}.Example: The set of even numbers between 1 and 10 can be written as {2, 4, 6, 8}.Set NotationRoster notation: Listing all the elements of a set within curly braces.Set-builder notation: Describing the set using a rule or condition.ExampleThe set of even numbers can be written in roster notation as {2, 4, 6, 8, ...}.The set of even numbers can also be written in set-builder notation as {x | x is an even integer}.Set-Builder NotationSet-builder notation is a concise way to define a set by specifying a rule or condition that its elements must satisfy. It involves using a curly bracket { }, a variable to represent elements of the set, a vertical line "|", and a condition that the elements must meet.General form:{x | condition(x)}This reads as "the set of all x such that condition(x) is true."Example:The set of even natural numbers can be written in set-builder notation as:{x | x is a natural number and x is even}The set of all real numbers greater than 3 can be written as:{x | x ∈ ℝ and x > 3}Set Operations/Defintions.Universal SetIn set theory, the universal set, often denoted by the symbol U, is the set that contains all the elements under consideration in a particular context. It's like the "master set" that encompasses all the possible elements related to a specific topic.Universal Set (U):{All colors}A = {red, orange, yellow, green, blue, indigo, violet} (set of colors in a rainbow)B = {black, white, gray} (set of neutral colors)C = {pink, purple, teal, brown} (set of pastel colors).Infinite SetsAn infinite set is a set that contains an unlimited number of elements. It cannot be counted to a finite number.Examples:The set of natural numbers: N = {1, 2, 3, ...}The set of integers: Z = {..., -2, -1, 0, 1, 2, ...}The set of real numbers: RFinite SetsA finite set is a set that contains a limited number of elements. It can be counted to a finite number.ExamplesThe set of vowels in the English alphabet: V = {a, e, i, o, u}The set of even numbers between 1 and 10: E = {2, 4, 6, 8, 10}Null Set (Empty Set)The null set or empty set, denoted by ∅ or {}, is a set that contains no elements.ExampleThe set of even prime numbers greater than 2 is empty.Singleton SetA singleton set is a set that contains exactly one element.ExampleThe set {5} is a singleton set.Elements and Members of a Set.Elements and members are synonymous terms used to describe the individual items that make up a set. They are the fundamental building blocks of a set.For example, if the set A is defined as {1, 2, 3}, then the elements or members of A are 1, 2, and 3.Elements of a set are usually enclosed in curly braces {}.To indicate that an element x belongs to a set A, we write x ∈ A.To indicate that an element x does not belong to a set A, we write x ∉ A.Example:If A = {1, 2, 3} and B = {2, 3, 4}, then:1 ∈ A (1 is an element of set A)4 ∈ B (4 is an element of set B)2 ∈ A and 2 ∈ B (2 is an element of both sets A and B)5 ∉ A (5 is not an element of set A)Subsets and SupersetsSubsets and supersets are relationships between sets that help us understand how sets are related to each other. A subset is a smaller set contained within a larger set, while a superset is a larger set that contains a smaller set.Subset:A set A is a subset of set B if all elements of A are also elements of B.In other words, every member of A is also a member of B.We denote this relationship as A ⊆ B.Example:If A = {1, 2} and B = {1, 2, 3}, then A is a subset of B because all elements of A (1 and 2) are also in B.Superset:A set B is a superset of set A if all elements of A are also elements of B.In other words, B contains all the elements of A and possibly more.We denote this relationship as B ⊇ AUsing the same example above, B is a superset of A because it contains all the elements of A (1 and 2) and an additional element (3). It is pertinent to note that;Every set is a subset of itself.The empty set is a subset of every set.If A is a subset of B and B is a subset of A, then A and B are equal sets.UnionThe union of two sets A and B, denoted as A ∪ B, is the set of elements that are in A or in B or in both.IntersectionThe intersection of two sets A and B, denoted as A ∩ B, is the set of elements that are in both A and B.ComplementThe complement of a set A, denoted as A', is the set of all elements in the universal set that are not in A.Difference:The difference of set A and set B, denoted as A - B, is the set of elements that are in A but not in B.ExamplesIf A = {1, 2, 3} and B = {2, 3, 4}, then:A ∪ B = {1, 2, 3, 4}A ∩ B = {2, 3}A' = {4, 5, 6, ...} (assuming the universal set is the set of natural numbers)A - B = {1}Cardinality of Sets.Cardinality is a fundamental concept in set theory that refers to the number of elements contained within a set. It provides a way to measure the size of a set.Examples:The set A = {1, 2, 3} has a finite cardinality of 3.The set of natural numbers N = {1, 2, 3, ...} has an infinite cardinality.The set of real numbers R has an uncountable infinite cardinality.Cardinality Notation:The cardinality of a set A is often denoted by |A|.For example, if A = {1, 2, 3}, then |A| = 3. EXERCISE 14.0.What is the intersection of the set of even numbers and the set of odd numbers?If A = {1, 2, 3} and B = {2, 3, 4}, what is A ∪ B?Given sets X = {a, b, c} and Y = {c, d, e}, find X ∪ Y.What is the union of the sets {1, 2, 3} and the empty set?If A = {1, 2, 3} and B = {2, 3, 4}, what is A ∩ B?Given sets X = {a, b, c} and Y = {c, d, e}, find X ∩ Y.If the universal set U = {1, 2, 3, 4, 5} and A = {1, 3, 5}, what is A'?Given set B = {a, b, c}, what is B' if the universal set is {a, b, c, d, e}?What is the complement of the set of all real numbers?Is the set {1, 2} a subset of the set {1, 2, 3}?If A = {a, b, c} and B = {a, b, c, d}, is A a subset of B?Can the empty set be a subset of any set?Given the sets A = {1, 2, 3} and B = {2, 3, 4}, find:A ∪ BA ∩ BA' (assuming the universal set is the set of natural numbers)A – BList the elements of the set {x | x is a prime number less than 10}.If A = {a, b, c} and B = {c, d, e}, find A ∪ B and A ∩ B.Submit Answers via Chat e.g Exercise 13.o, then type Answers (Number your Answers).FOR MORE CONTENT!!!Register HERE Now! Pick a course, watch our Videos and take our CBT's.Additional Resources:N/A: https://www.waokmath.com[Image Source]: https://www.freepik.com/.
