I.2 The Language and Grammer of Mathematics, 2. Four Basic Concepts and Pages 9 – 12

The word “is” and its uses: 2 is the square root of four, 2 is less than 5 and 2 is an even prime number.   These simple sentences are used to elucidate the concepts of sets, functions, relations and binary operations.

We define a collection of objects to be a set.   Thereby we can discuss these collections whether finite or infinite.  Sets are used to define mathematical objects and in proving statements about mathematical reasoning.  “Is an element of” is the mathematical symbol “Γ.

Functions are defined as the process of transforming objects.  f(x) = y means the function “f” transforms the object “x” into the object “y”.  Functions have properties and there exist sets of functions, for example, Hilbert Spaces, Function Spaces and Vector Spaces.  Some functions can be inverted.  Functions have a domain and a range.  They transform objects from the domain into objects in the range.  “f: A ® B means “f” is a function with domain “A” and range “B” with “:” read as “such that”.  The image of the function “f” is the set of values actually taken by f(x) for xÎA.  The word “image” is also used as the one element f(x).

One way of writing f(x) = y is f: x # y.  Notice this “bar arrow” is for elements of sets, whereas above, the arrow without the bar is used on the sets.

An “injection” means that f(x) and f(y) are different whenever x and y are different.  

A “surjection” means that every element y of the image B is equal to f(x) for some element x of the domain A.  So, there is a function”g” where g(y) = x.

A function that is both an injection and a surjection is called a “bijection”.  A function that is a bijection has an inverse (and visa versa). 

“Is less than” is denoted by the symbol “<“.  Three examples of relations are: =, < and Î.  Strictly speaking the relation “<” in the sentence “a < b” is different if “a” and “b” are members of the positive integers than if “a” and “b” are members of the reals.  We ususally don’t even see this difference.

Sometimes relationships are between two sets.  If A is the set of positive integers and B the set of all sets of positive integers then the relationship “Δ is between the two sets, that is, A Î B. 

“Equivalence relations” regard objects as “essentially the same”.  Examples are “is similar to” and “is conguent modulo m to”.  “Similar” refers to geometric objects that can be transformed into one another by a combination of reflections, rotations, translations, and enlargements.  Two numbers that differ by a multiple of m are said to be conguent modulo m.

Relations take a set and divide it into equivalence classes where each part consists of objects that are essentially the same.

Define a relation on a set A as “~”.  An equivalence relation has the following three properties: reflexive, symmetric and transitive.  Reflexive means x ~ x for all x  Î A.  Symmetric means if x ΠA and y Î A and x ~ y then it must also be that y ~ x.  Transitive means that x ΠA, y ΠA and z Î A such that x ~ y and y ~ z then it must be true that x ~ z. 

The relations “is similar to” and “is conguent to modulo m” both have the properties of being reflexive, symmetric and transitive.  The relation “is less than” is transitive but not reflexive or symmetric.

Equivalence relations are used to make precise the notion of a quotient.

Examples of “binary operations” are: “plus”, “minus”, “times”, “divided by” and “raised to a power”.   A binary operation is a function from the set of pairs (x,y), where x Î A and y Î A, of domain A, onto the range A.  However, interestingly, we write x + y rather than +(x,y).

A binary operation may have the properties: commutative, associative, identity and inverse.  Let “*” represent a binary operation.  Commutative means x * y = y * x.  Associative means (x * y)  * z = x * (y * z).  By convention binary operations need to be defined for all x Î A.

An element e Î A is called an identity if e * x = x * e = x for all x Î A.   Examples are 0 and 1 for plus and times.

If * has an identity e and x Î A then an inverse for x is an element y Î A such that x * y = y * x = e.   Examples for plus and times are: -x and 1/x.

The basic properties of binary operations are fundamental to the structures of abstract algebra.

                      

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