Sets

Introduction

Useful Logic

Logic 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 Connectives

In order to analyse a problem logically, a precise language must be used.

Propositions

The 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 Connectives

In 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↔⇔, ≡

Arguments

Axiom of Extensionality

Basically, 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

𝑨=𝑩

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

𝑨≠𝑩

Axiom of ExtensionalityTwo sets are equal if and only if they have the same elements.

Sources and References

• https://en.wikipedia.org/wiki/Set_(mathematics)
• https://en.wikipedia.org/wiki/Set_theory
• https://en.wikipedia.org/wiki/Axiom_of_extensionality
• https://en.wikipedia.org/wiki/Extensionality
• https://en.wikipedia.org/wiki/Logic
• Irving M. Copi, Carl Cohen, 2014, Introduction to Logic, Pearson Education Limited, USA
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