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SetsIntroductionIn general, a set is a collection of objects. However, mathematically, a set is a mathematical model for representing a welldefined unordered collection of distinct objects. For example,The collection of numbers, 9, 1, 5, 7, 3, 1, 3, can be viewed as a set that contains the five distinct elements 1, 3, 5, 7, 9.
A set is usually denoted by a capital letter, such as 𝑨, 𝑩, 𝑪, …, 𝑿, 𝒀, 𝒁, while lowercase letters, such as 𝑎, 𝑏, 𝑐, …, 𝑥, 𝑦, 𝑧, are used as symbols to denote unspecified objects of sets for a statement. For example,
Let 𝑨 be the set that contains the five distinct elements 1, 3, 5, 7, 9.
Let 𝑨 be the set of all elements 𝑎 is an odd natural numbers and 𝑎 is less than 10.
In other words, numbers, 1, 3, 5, 7, 9 are elements or members of set 𝑨. The term "is element of" is equivalent to "belongs to" or "is in". And the statement can be written as
While numbers 2, 4, 6, 8 are not elements or members of the set 𝑨. The term "not element of" is equivalent to "not belongs to" or "not in". And the statement can be written as
The typical ways to define a set are roster form, semantic description and setbuilder form.
Special Sets of NumbersSome special sets of numbers areLet ℕ be the set of all natural numbers. ℕ = {1, 2, 3, …}, but sometimes ℕ may contain number 0.
Useful LogicLogic is a way of thinking by inferring with the concept of correct reasoning. In other words, an output, either making a guess or forming an opinion, is produced based on using the speicified information as input. In general, logic is used to distinguish sound and faulty reasoning.Propositions and Sentential ConnectivesIn order to analyse a problem logically, a precise language must be used.PropositionsThe precise building block used in logic reasoning is called proposition. A proposition is a declarative statement which is either true or false, but not both.Sentential ConnectivesIn order to express mathematical statements to symbolic logical forms, some sentential connectives or logical connectives are defined. Keywords: 'not', 'and', 'or', 'if …, then …', 'if and only if' are the sentential connectives used in sentential logic. The logical connectives of sentential logic are Logic ConnectivesSentential ConnectivesSentential Logic SymbolRemarks NegativeNot¬∼, ! ConjunctionAnd∧., & DisjunctionInclusive Or∨+, ∥ Implication, Conditionalif …, then …→⇒ Double Implication, Biconditional, Equivalenceif and only if↔⇔, ≡ArgumentsAxiom of ExtensionalityBasically, the foundation of set theory is based on the concept of belonging. In other words, a set is only used to represent the unordered elements contained in the set as in roster form. Both semantic description and set builder forms are only used to specify the elements in the set. However, rules used to determine the memners of the set may also be important properties of the elements of the set. Therefore, if the members of set 𝑨 is the same as the members of set 𝑩, set 𝑨 and set 𝑩 are equal. The equality of two sets A and B can be denoted as
However, if the members of set 𝑨 is not the same as the members of set 𝑩, set 𝑨 and set 𝑩 are not equal. And can be expressed as
Axiom of ExtensionalityTwo sets are equal if and only if they have the same elements.
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