Sets & Whole-Number Operations and Properties

In the addition of whole numbers, if A and B are two disjoint sets, and n(A) = a and n(B) = b, then a+b=n(AUB).In the equation a+b=c, a and b are addends and c is the sum

Closure property of addition=for whole numbers a and b, a+b is a unique whole numberIdentity property of addtion=there exists a unique whole number, 0, such that 0+a=a+0 for every whole number a. Zero is the additive identity element.Commutative property of addition=for whole number a and b, a+b=b+aAssociative property of addition=for whole numbers a,b, and c, (a+b)+c=a+(b+c)

Closure property of addition=for whole numbers a and b, a+b is a unique whole number

Identity property of addtion=there exists a unique whole number, 0, such that 0+a=a+0 for every whole number a. Zero is the additive identity element.

Commutative property of addition=for whole number a and b, a+b=b+a

Associative property of addition=for whole numbers a,b, and c, (a+b)+c=a+(b+c)

In the subtraction of the whole numbers a and b, a-b=c if and only if c is a unique whole number such that c+b=a. In the equation a-b=c, a is the minuend, b is the subtrahend, and c is the difference.

Even though subtraction of whole numbers is closely related to addition of whole numbers, the properties of addition do not hold for subtraction.

Multiplication as repeated addition=if there are m sets with n objects in each set, then the total number of objects (n+n+n+...+n, where n is used as an addend m times) can be represented by m x n, where m and n are factors and m x n is the productMultiplication of whole numbers=if A and B are finite sets with a = n(A) and b = n(B), then a x b = n(A x B). In the equation a x b = n(A x B), a and b are called factors and n(A x B) is called the product

Closure property of multiplication=for whole number a and b, a x b is a unique whole numberIdentity property of multiplication=there exists a unique whole number, 1, such that 1 x a = a x 1 = a for every whole number a. Thus 1 is the multiplicative identity elementCommutative property of multiplication=for whole number a and b, a x b = b x aAssociative property of multiplication=for whole numbers a, b, and c, (a x b) x c = a x (b x c)Zero property of multiplication=for each whole number a, a x 0=0 x a=0Distributive property of multiplication over addition=for whole numbers a, b, and c, a x (b+c)=(a x b) + (a x c)

Closure property of multiplication=for whole number a and b, a x b is a unique whole number

Identity property of multiplication=there exists a unique whole number, 1, such that 1 x a = a x 1 = a for every whole number a. Thus 1 is the multiplicative identity element

Commutative property of multiplication=for whole number a and b, a x b = b x a

Associative property of multiplication=for whole numbers a, b, and c, (a x b) x c = a x (b x c)

Zero property of multiplication=for each whole number a, a x 0=0 x a=0

Distributive property of multiplication over addition=for whole numbers a, b, and c, a x (b+c)=(a x b) + (a x c)

Set=any collection of objects or ideas that can be listed or describedElement=each individual object in a setOne-to-one correspondence=Sets A & B have one-to-one correspondence if and only if each element of A is paired with exactly one element of B and each element of B is paired with exactly one element of AEqual sets=Sets A & B are equal sets if and only if each element of A is also an element of B and each element of B is also an element of A (symbolized A=B)Equivelent sets=Sets A & B are equivalent sets if and only if there is a one-to-one correspondence between A and B (symbolized by A ~ B)Subset=For all sets A & B, A is a subset of B if and only if each element of A is also an element of B (symbolized as A C B)Proper Subset=For all sets A & B, A is a proper subset of B if and only if A is a subset of B and there is at least one element of B that is not an element of A (symbolized as A C B)Complement=The complement of set A consists of all of the elements in U that are not in AUnion of 2 sets=A and B is the set containing every element belonging to set A or set B and is written A U BIntersection of 2 sets=A and B is the set containing every element belonging to both set A and set BDisjoint sets=if and only if their intersection is the empty set

Whole number=the unique characteristic embodied in each finite set and all the sets equivalent to itCardinality=the number of elements in set A is expressed as n(A)Less than is symbolized as <Greater than is symbolized as >Set of natural numbers=infinite set, proper subset of whole numbers, also called the set of counting numbers, does not include 0Set of even numbers=a proper subset of the set of whole numbers starting with 0 and continuing with every second number after thatSet of odd numbers=proper subset of the whole numbers that remains after the even numbers are removed

In the division of whole numbers a and b, b does not equal 0, a/b=c if and only if c is a unique whole number such that c x b=a. In the equation, a/b=c, a is the dividend, b is the divisor, and c is the quotient.

As with multiplication, division of whole numbers would have the closure property provided that when we divide two whole numbers, the quotient is a unique whole number.