This page contains definitions of terms which students may not have heard of or may forget what they are. For the most part these are covered in other math courses (such as MATH 1046 or MATH 1056) or should have been covered in highschool.

The definitions included here are not necessarily complete or written in correct mathematical form. Instead they have been simplified to give a general idea of the concept while minimizing the confusion often associated with complete mathematical definitions. Therefore the definitions should be used only to help understand a concept not as a resource for mathematically correct definitions.

An excellent example of this is the definition of a relation. This definition is incorrect to the extent that if you were to tell it to your professor he/she would probably faint. However the actual definition of a relation is far to complex to include in this glossary so the definition provided is meant only to give the basics of a relation as you may need in the MATH 1036/1037 course.

INDEX

• Antiderivative
• Arbitrary Number
• Axiom
• Complex Numbers
• Conditional Statement
• Contrapositive
• Converse
• Counterexample
• Domain
• Even Function
• Explicit Function
• Function
• If and only if (IFF)
• Implicit Function
• Inverse
• Horizontal Asymptote
• Horizontal Line Test
• Lemma
• Math Amnesia
• Mean Value Theorem
• Odd Function
• One-to-One
• Oscillating Function
• Periodic Function
• Proof by Contradiction
• Proof by Induction
• Range
• Relation
• Rolle's Theorem
• Slant Asymptote
• Vertical Asymptote
• Vertical Line Test

Antiderivative

Antiderivative of a function f(x) is another function derivative of which is equal to f(x). Thus, the process of finding antiderivative is inverse to that of finding derivative. Antidifferentiation is the process by which differentiation (taking the derivative) is reversed and antiderivative is found. Antiderivative is also called an indefinite integral.

Arbitrary Number

A number which could be any number it is defined to be but for which no specific value is chosen. It is often used in proofs since it can represent any number but does actually have the value of any number so that the proof applies to more than one situation.

Axiom

A statement that is so self-evident no proof is needed.

Complex Numbers

A number consisting of a two parts: a real part and an imaginary part. Commonly expressed in the form a+bi where a and b are real numbers and i has the property that i²=-1. Thus a is the real part and bi is the imaginary part. This allows the number to be graphed in the complex plane (Argand plane) with coordinates (a,b).

Conditional Statement

A logical statement of the form "if this then that" meaning that "if this" happens "then that" will happen. Commonly referred to as an if-then statement.

Contrapositive

A statement of the where hypothesis and conclusion are reversed and negated. I.E. the conditional statement "If this then that" becomes "If not that then not this". The contrapositive is equivalent to the original statement.

Converse

A statement where the hypothesis and conclusion are reversed. I.E. the conditional statement "If this then that" becomes "If that then this". The Converse is NOT equivalent to the original statement, but is equivalent to the inverse.

Counterexample

A specific example used to disprove something. For example if someone were to say "when you multiply two numbers the answer is always even", a counterexample for this would be 1x3=3 since it shows that two numbers can be multiplied can have an odd answer. Most of the errors in the Common Mistakes section are shown using counterexamples.

Domain

The domain of a function is the set of all values of a variable  for which the  function is defined. For example, the domain of the function y=x² is the whole real line since y has a value for every real value of x whereas the domain of the function y=(1 / x) x0 (the whole real line except x=0) since y is undefined when x=0.

Even Function

A function with the property that f(-x)=f(x). This means that when -x is substituted into f for x the function stays the same. Functions of this type are symmetric about the y-axis meaning that the graph of f(x) for x0 is a reflection of the graph of f(x) for x0. An example of this is the function f(x)=x² since f(-x)=(-x)²=x²=f(x).

Explicit Function

A function where one variable is explicitly (e.g. by means of formula or table of values) expressed in terms of another. Examples are y=2x+2, q=3p²-2p-30, and s=3sin t.

Function

A rule where each value from the domain set is assigned to exactly one element in the range set. A common test for a function is the vertical line test where on a graph if any vertical line intersects the graph more than once it is not a graph of a function.

If and only if (IFF)

An expression used to imply that a statement holds in both directions and only in the described situations. This means that if you have the situation described on either side of the 'if and only if ' then you will have the situation on the other side as well and if you do not have one then you will not have the other. An if and only if statement is also called a biconditional statement.

Implicit Function

A function where one variable is not expressed explicitly in terms of another, but where it is still assumed that one variable depends on another. Examples are y+x=2, pq²=2p-3q, and s/t=2t

Inverse

A statement where the hypothesis and conclusion are negated. I.E. the conditional statement "If this then that" becomes "If not this then not that". The inverse is NOT equivalent to the original statement, but is equivalent to the converse.

Horizontal Asymptote

A horizontal line which the graph of a function approaches as variable tends to positive or negative infinity. It should be noted that the graph can cross the horizontal asymptote as many times as it likes (as with many oscillating functions). A horizontal asymptote occurs when the limit of a function as the variable approaches either positive or negative infinity is a constant.

Horizontal Line Test

A technique used to test if a function is one-to-one where if any horizontal line drawn on a graph intersects the graph of the function more than once that function is not one-to-one.

Lemma

Often referred to as a 'mini theorem'. A Lemma is a fact which must be proved so it can be used in the proof of another theorem.

Math Amnesia

The term given to describe the sudden loss of mathematical knowledge. This commonly occurs at high stress times such as tests and exams and is often blamed for poor scores received on them. Math Amnesia can be overcome by stress reducing techniques and the use of better preparation techniques. See Test/Exam Preparation strategies in the Study Tips tutorial for suggestions on how to prevent Math Amnesia.

Mean Value Theorem

A theorem which states that if a function f is continuous on [a,b] and differentiable on (a,b) then there is a number c in (a,b) that satisfies the property f '(c)=(f(b)-f(a)) / (b-a). In words the theorem says that if the function is continuous on the closed interval and differentiable on the open interval then there is a point in the open interval such that the slope of the tangent at that point is equal to the slope of the secant joining the endpoints of the interval.  I.E. there is a point where the slope of the tangent equals the slope of the secant or the instantaneous rate of change is equal to the average rate of change. Geometrically, this means that there is a point on the graph where the tangent line is parallel to the secant line. Note that there could be more than one number in the interval with this property.

Odd Function

A function with the property that f(-x)=-f(x). This means that when -x is substituted into f for x the sign of the function changes. Functions of this type are symmetric about the origin:  the graph of function for x0 appears as the graph of function for x0 rotated 180 degrees around the origin. An example of this is the function f(x)=x³ since f(-x)=(-x)³=-(x³)=-f(x).

One-to-One

A function is one-to-one (also called injective) if, and only if, it has the property that if f(x₁)=f(x₂) then x₁=x₂. In other words it never takes on the same value twice. A quick test for this property is the horizontal line test where if a horizontal line intersects the graph of a function more than once the function is not one-to-one.

Oscillating Function

A function whose graph continuously switches between increasing and decreasing causing the graph to have a series of local maxima and minima resembling waves in water or a vibrating string.

Periodic Function

A function f is said to be periodic if f(x+p)=f(x) for all x in the domain of f where p is the period (smallest positive number for which this property holds). In other words the graph of function repeats itself indefinitely. An example of this is f(x)=sin(x) where the period is 2 since sin(x+2)=sin(x) for all x.

Proof by Contradiction

A type of proof whereby the opposite of what is being proved is assumed to be true and a sequence of statements is obtained until a contradiction (an impossible scenario) is reached thus indicating that the assumption is false, so the original statement must be true.

Proof by Induction

A type of proof where the following series of steps is followed. The statement is proved true for one specific integer value (usually 0, 1 or 2). This is called the base of induction. The statement is then assumed to be true for some arbitrary value (which is a positive integer). This is called the inductive assumption. A proof is then formulated to prove the statement true for the arbitrary value +1 making use of the assumption that the statement is true for the arbitrary value. This is called inductive step. It is then concluded that since the statement is true for an arbitrary value +1 given that it is true for an arbitrary value it should be true for any positive integer. Therefore the entire statement is proved. This type of proof can only be used for number sets who increase and decrease by integer values I.E. the set of whole numbers, natural numbers, positive integers, non-negative integers, negative integers, etc. For more information, see the Induction tutorial.

Range

The range of a function is the set of values the function takes as the variable goes through all the values of the domain. For example the range of the function y=x³ is the whole real line since all the values in the real line have corresponding values in the domain that the function takes them to whereas the range of y=x² is y0 (any value greater than or equal to 0) since the function does not take any values below y=0 for any value in the domain.

Relation

A relation (or, more precisely, a binary relation), is a condition on two numbers that is either satisfied for a given pair of numbers, or is not satisfied. For example, ">" is a relation. A relation between two real numbers can be viewed as a subset of a coordinate plane. For example, relation y > x distinguishes the region strictly above the line y = x.

Rolle's Theorem

A theorem which states that if a function f is continuous on [a,b], is differentiable on (a,b), and f(a)=f(b) then there will be a point c in (a,b) with the property that f '(c)=0. In words the theorem says that if the function is continuous on a closed interval, differentiable on the open interval, and the value of the function at the endpoints of the interval are equal, then there is a point in the open interval such that the slope of the tangent line at that point is 0. I.E. There is a local maximum or minimum on the interval or the function is constant. Geometrically, Rolle's Theorem states that there is a point on the graph where the tangent line is horizontal. Note that there could be more than one number in the interval with this property.

Slant Asymptote

An oblique line (neither horizontal or vertical) which the graph of a function approaches as the variable goes to positive or negative infinity. Often occurs in rational functions when the degree of the numerator is one higher than the degree of the denominator.

Vertical Asymptote

A vertical line which the graph of the of a function approaches but never reaches. Occurs when the limit of a function as it approaches a number (from the left, right, or at the number) equals either positive or negative infinity.

Vertical Line Test

A technique used to determine whether a curve in coordinate plane is a function or not where if a vertical line crosses the graph more than once that relation is not a function.

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