Principles of Modern Mathematics Exam 1 terms

Carthage College

Natural Numbers

The set of positive integers starting from 1 and extending indefinitely. Natural numbers are used for counting and ordering.

Integers

The set of whole numbers that include both positive and negative numbers, as well as zero. Integers are used in various mathematical computations and concepts.

Rational numbers

The set of numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Rational numbers include fractions and whole numbers.

Irrational Numbers

Numbers that cannot be expressed as a fraction of two integers, typically represented by non-repeating, non-terminating decimals. Examples include (\sqrt{2}) and (\pi).

Real numbers

The set of all numbers that includes both rational and irrational numbers, encompassing integers, fractions, and non-repeating decimals. Real numbers can be represented on a number line.

Sets

A collection of distinct objects, considered as an object in its own right. Sets can contain numbers, symbols, or even other sets.

Elements of a set

are the distinct objects that belong to that set.

cardinality of a set

is the measure of the number of elements in a set.

union of 2 or more sets

is the set containing all elements from the involved sets, without duplicates.

not (-)

a condition that excludes certain elements from a set.

Cartesian product

is the set of all ordered pairs made from elements of two sets, combining each element of the first set with each element of the second.

intersection of 2 or more sets

is the set containing all elements that are common to the involved sets.

compliment of a set

is the set of all elements not in the given set, relative to a universal set.

universal set

set that contains all the elements of multiple sets

empty/null set

set that has no elements

Hindu-Arabic system

base 10 number system that a majority of the world uses

face value in a number system

The value of a digit based on its position in a number, irrespective of its location. For example, in the number 543, the face value of 5 is 5.

place value in a number system

The value of a digit based on its position within a number, affecting its total contribution to the number's overall value. For example, in the number 543, the place value of 5 is 500.

prime numbers

Natural numbers greater than 1 that have no positive divisors other than 1 and themselves.

composite numbers

Natural numbers greater than 1 that are not prime, meaning they have divisors other than 1 and themselves.

Fundamental Theorem of arithmetic

States that every integer greater than 1 can be uniquely factored into prime numbers.

greatest common divisor/factor

The largest positive integer that divides two or more integers without leaving a remainder.

Least common multiple

The smallest positive integer that is a multiple of two or more integers.

Group

A set equipped with an operation that combines any two elements to form a third element, satisfying four properties: closure, associativity, identity element, and inverses.

Identity

element in a group that leaves other elements unchanged when combined.

Closure

A property of a set under an operation where the result of the operation on any two elements in the set is also an element of the set.

associativity

If a and b are elements of G, then (a*b) * c = a * (b * c) for any elements a, b, and c in G.

Inverse of an elements

An element in a group that, when combined with a given element, results in the identity element of the group.

commutativity of a set

A property of a set under an operation where the order of the elements does not affect the result, meaning a * b = b * a for any elements a and b in the set.