Week 3: Mathematical Language and Symbols

• Mathematics as a language has symbols to express a formula or represent a constant.
• It has syntax to make the expression well-formed to make the characters and symbols clear and valid that do not violate the rules.
• An expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.
• A mathematical sentence is the analogue of the English sentence; it is a

  correct arrangement of mathematical symbols that states a complete thought.
• A mathematical convention is a fact, name, notation, or usage which is generally agreed upon by mathematicians.
• Order of operations is the hierarchy of mathematical operations.
• Brackets - These act as assort of override from the rest of the order of operations.
• Order - Anything to the power of anything else goes next. This tends to help

  to make complicated expressions took a bit neater;
• BODMAS/PEMDAS - order of operations
• 4 Basic Concepts of Mathematics

  • Set
  • Relation
  • Function
  • Binary Operation
• Set is a collection of well-defined objects that contains no duplicates.
• The objects in the set are called the elements of the set.
• A finite set contains elements that can be counted and terminates at certain

  natural number, otherwise, it is infinite set.
• The empty set, or null set, ∅ or { }, which has no members at all
• A set with only one member is called a singleton or a singleton set
• List Notation/Roster Method – by listing all its members

  Examples:

  1. {1, 12, 45}{George Washington, Bill Clinton}
  2. {George Washington, Bill Clinton}
• Predicate Notation/Rule – by stating a property of its elements. It has a property that the members of the set share (a condition or a predicate which holds for members of this set).
• Recursive Rules – by defining a set of rules which generates or defines its members
• Equal Sets - Two sets are equal if they contain exactly the same elements.
• Two sets are equivalent if they contain the same number of elements.
• Universal Set - A set that contains all the elements considered in a particular situation and denoted by U.
• Subsets - A set A is a subset of set B if every element of A is also an element of B, “A is a subset of B” is written as A ⊆ B.
• Proper subset is a subset that is not equal to the original set, otherwise, improper subset.
• Cardinality of the Set - It is the number of distinct elements belonging to a finite set.
• Power Set - It is the family of all the subsets
• Union is an operation for sets A and B in which a set is formed that consists of all the elements included in A or B or both denoted by ∪ as A ∪ B.
• Intersection is the set containing all elements common to both A and B, denoted by ∩.
• Complementation is an operation on a set that must be performed in reference to a universal set, denoted by A’.
• Relation is a rule that pairs each element in one set, called the domain, with one or more elements from a second set called the range. It creates a set of ordered pairs.
• Function is a rule that pairs each element in one set, called the domain, with exactly one element from a second set, called the range.
• A binary operation on a set is a calculation involving two elements of the set to produce another element of the set.
• A proposition (or statement) is a sentence that is either true or false (without additional information.
• A denial is a statement equivalent to the negation of a statement.
• A tautology is a statement which is always true
• A contradiction is a statement which is always false.
• A predicate (open sentence) is a sentence containing one or more variables which becomes a proposition upon replacement of the variables.