Math 30-2 Diploma key terms

A compound statement formed by joining two statements with the word 'and'. It is true only when both of the original statements are true; otherwise it is false.

A compound statement formed by joining two statements with the word 'or'. It is false only when both original statements are false; if at least one part is true, the whole is true.

The opposite of a statement, usually formed by adding 'not' or by rewording so the statement's truth value flips. If a statement is true, its negation is false, and vice versa.

A statement of the form 'if P then Q', where P is the hypothesis and Q is the conclusion. It is false only in the case where P is true but Q is false.

The statement formed by swapping the hypothesis and conclusion of a conditional. 'If P then Q' becomes 'if Q then P'; the converse is not logically equivalent to the original.

The statement formed by negating both the hypothesis and conclusion of a conditional. 'If P then Q' becomes 'if not P then not Q'; it is not equivalent to the original statement.

The statement formed by swapping and negating both parts of a conditional. 'If P then Q' becomes 'if not Q then not P'; it is always logically equivalent to the original.

A table that lists every possible combination of truth values for the simple statements in a compound statement and shows the resulting truth value of the whole.

The classification of a statement as either true or false. Every well-formed logical statement has exactly one truth value.

Two statements are logically equivalent when they have identical truth values in every possible case, so one can replace the other without changing meaning.

A reasoning structure in which the conclusion must be true whenever all the premises are true. Validity concerns the form of the argument, not the actual truth of the premises.

A statement offered as a reason or piece of evidence in an argument. The conclusion is supposed to follow from the premises.

A single specific case that shows a general statement or argument is false. Finding one counterexample is enough to disprove a universal claim.

A general statement believed to be true based on observed patterns but not yet proven. It can be tested by searching for counterexamples.

Reasoning that moves from specific observations or examples to a general conclusion. The conclusion is probable but not guaranteed, so it can be overturned by a counterexample.

Reasoning that moves from general statements accepted as true to a specific conclusion that must follow. If the premises are true and the form is valid, the conclusion is certain.

An arrangement of items in which the order matters. The number of ways to arrange r items chosen from n distinct items is found using factorial expressions.

A selection of items in which the order does not matter. Choosing a group of 3 from 10 people is a combination because rearranging the same 3 people gives the same group.

The product of all positive whole numbers from 1 up to a given number, written with an exclamation mark. For example, 4! equals 4 × 3 × 2 × 1 = 24, and 0! is defined as 1.

If one choice can be made in m ways and a second independent choice in n ways, then the two choices together can be made in m × n ways.

Two events that cannot both happen at the same time. For such events, the probability of either occurring is the sum of their individual probabilities.

Two events that can occur together in at least one outcome. When finding the probability of either, the overlap counted twice must be subtracted once.

A probability rule for finding the chance that event A or event B occurs. For overlapping events it is P(A) + P(B) − P(A and B).

A probability rule for finding the chance that two events both occur. For independent events it is the product of their individual probabilities.

Two events where the outcome of one has no effect on the probability of the other. The probability of both happening equals the product of their separate probabilities.

Two events where the outcome of one changes the probability of the other. The second probability must be recalculated based on what already happened.

The probability that an event occurs given that another event has already occurred. It is written P(A|B) and read as the probability of A given B.

The complete set of all possible outcomes of a probability experiment. Listing the sample space helps count favourable outcomes and total outcomes.

The probability calculated from counting outcomes, found by dividing the number of favourable outcomes by the total number of equally likely outcomes in the sample space.

The probability estimated from actually performing an experiment, found by dividing the number of times an event occurred by the total number of trials.

A relation in which each input value is paired with exactly one output value. Passing the vertical line test confirms a graph represents a function.

Any set of ordered pairs linking input values to output values. A relation may or may not be a function depending on whether any input repeats with different outputs.

The set of all possible input values (usually x-values) for which a relation or function is defined. It describes what values may be put into the rule.

The set of all possible output values (usually y-values) a function actually produces. It describes the results that come out of the rule.

A visual check for functions: if no vertical line crosses a graph more than once, the graph represents a function because no input has two outputs.

A visual check for one-to-one behaviour: if no horizontal line crosses a graph more than once, each output has only one input, so the inverse is also a function.

A function that reverses the effect of another, swapping inputs and outputs. Its graph is a reflection of the original graph across the line y = x.

A line that a graph approaches more and more closely but generally never touches or crosses. Exponential and rational functions commonly have horizontal or vertical asymptotes.

A function of the form y = a·b^x where the variable is in the exponent and the base b is positive and not equal to 1. It models growth when b > 1 and decay when 0 < b < 1.

The exponent to which a base must be raised to produce a given number. It is the inverse of an exponential relationship, so it can solve equations with the unknown in an exponent.

Half the vertical distance between the maximum and minimum values of a periodic graph. It measures how far the wave rises or falls from its middle level.

The length of the smallest repeating section of a periodic graph, or the horizontal distance for one complete cycle. It tells how often the pattern repeats.

The horizontal centre line midway between the maximum and minimum values of a sinusoidal graph. It is the vertical shift of the wave from the x-axis.

A function whose graph has the smooth repeating wave shape of sine or cosine, described by amplitude, period, phase shift, and vertical displacement.

The horizontal translation of a sinusoidal graph from its usual starting position. It tells how far left or right the wave has been moved.

The acute angle between the terminal arm of an angle and the nearest part of the x-axis. It helps determine trigonometric values in any quadrant.

An angle drawn on a coordinate plane with its vertex at the origin and its initial arm along the positive x-axis. The angle is measured from that starting arm.

The rotating ray of an angle in standard position that determines where the angle ends. Its position identifies the quadrant and the signs of the trigonometric ratios.

A circle of radius 1 centred at the origin, used to define sine and cosine as the y- and x-coordinates of points on the circle. It links angles to trigonometric values.

In a right triangle, the ratio of the side opposite a given angle to the hypotenuse. On the unit circle it equals the y-coordinate of the point at that angle.

In a right triangle, the ratio of the side adjacent to a given angle to the hypotenuse. On the unit circle it equals the x-coordinate of the point at that angle.

In a right triangle, the ratio of the side opposite a given angle to the side adjacent to it, equal to sine divided by cosine. It can be undefined where cosine is zero.

A rule stating that in any triangle the ratio of a side to the sine of its opposite angle is constant for all three sides. Useful for non-right triangles when a matching pair is known.

A rule relating the three sides of a triangle to one angle, generalizing the Pythagorean theorem. It is used to solve non-right triangles when three sides or two sides and the included angle are known.