Free Math 30-2 Diploma Exam Flashcards

A conjecture is a statement believed to be true based on observations or patterns. It still needs proof or a valid counterexample test.

One valid counterexample disproves a conjecture that claims something is always true. More supporting examples do not repair the false claim.

Inductive reasoning builds a general pattern from examples. Deductive reasoning starts from accepted facts or rules and reaches a logically necessary conclusion.

A' is the complement of A: all elements in the universal set that are not in A. The universal set must be known before a complement is meaningful.

Union means elements in A or B or both. Intersection means elements in both A and B at the same time.

Disjoint sets have no elements in common, so their intersection is empty. In a Venn diagram, their circles do not overlap.

Every element of A is also an element of B. A can be equal to B unless the statement says A is a proper subset of B.

It has 2^n subsets because each element has two choices: included or not included. This count includes the empty set and the whole set.

Add the number in A, add the number in B, then subtract the overlap once: n(A union B) = n(A) + n(B) - n(A intersect B).

It separates only A, only B, both, and neither. Counting each region once keeps overlap from being added twice.

Use it when a process has stages with independent choices. Multiply the number of choices for each stage to count total outcomes.

Ask whether order matters. If order creates a different outcome, use a permutation; if the same group counts once, use a combination.

nPr counts ordered arrangements of r objects chosen from n distinct objects. It is used for rankings, positions, or roles where order matters.

nCr counts unordered selections of r objects from n distinct objects. It is used for committees, groups, or choices where order does not matter.

Divide by factorials for repeated items. For n total objects with repeats of a, b, and c, distinct arrangements are n!/(a!b!c!).

Rotations of the same seating are not new arrangements, so n distinct people around a circle can be arranged in (n - 1)! ways when only rotations match.

P(not A) = 1 - P(A). It is especially useful for at least one questions, where counting no successes may be easier.

Probability compares favourable outcomes to total outcomes. Odds in favour compare favourable outcomes to unfavourable outcomes.

Use a/(a + b), because a outcomes are favourable and b are unfavourable, making a + b total outcomes in the ratio model.

They cannot occur on the same trial. If A and B are mutually exclusive, P(A and B) = 0 and P(A or B) = P(A) + P(B).

The overlap must be subtracted once: P(A or B) = P(A) + P(B) - P(A and B). Otherwise the overlap is counted twice.

A and B are independent when one event occurring does not change the probability of the other. Algebraically, P(A and B) = P(A)P(B).

The first draw changes the sample space for the second draw. Update both the favourable count and the total count before multiplying.

It means the probability of B given that A has already occurred. The sample space is restricted to outcomes where A is true.

Use the given condition as the denominator. For P(B|A), look only within the A row, column, or group, then count how many of those also have B.

Use the complement: 1 - P(no successes). This is usually faster than adding many separate success cases.

The degree is the highest exponent with a nonzero coefficient. It helps predict end behaviour and the maximum possible number of x-intercepts.

Together with degree, it describes what happens as x becomes very large positive or negative. Even and odd degrees behave differently at the two ends.

Zeros are x-values where f(x) = 0. On a graph they are x-intercepts when the graph crosses or touches the x-axis.

In y = a(x - h)^2 + k, the vertex is (h,k). If a is positive, the vertex is a minimum; if a is negative, it is a maximum.

Set x = 0 and evaluate the function. The y-intercept is the output where the graph crosses the y-axis.

An exponential function gives a power from an exponent, while a logarithm returns the exponent needed to produce a value with the same base.

The base must be positive and not equal to 1, and the argument must be positive. These restrictions define the domain of logarithmic functions.

A growth model has multiplier greater than 1, such as A = a(b)^t with b > 1. Each equal time step multiplies by the same factor.

A decay model has multiplier between 0 and 1, such as A = a(b)^t with 0 < b < 1. The amount decreases by a constant percent each step.

Each half-life multiplies the amount by 1/2. After n half-lives, the remaining amount is initial amount times (1/2)^n.

In A = P(1 + r)^t for annual compounding, 1 + r is the yearly multiplier. It includes the original 100% plus the interest rate.

Write b^c = a. This rewrite is often the fastest way to evaluate logarithms or solve simple logarithmic equations.

Without vertical shifts, the horizontal asymptote is y = 0. The graph approaches zero but does not cross it in the basic model.

Amplitude is half the distance between maximum and minimum. The midline is the average of maximum and minimum and represents the vertical centre.

The period is the input length required for one complete cycle before the pattern repeats.

Frequency is cycles per unit time, so frequency = 1/period when period is measured in that time unit.

It moves the entire graph up or down and changes the midline, but it does not change amplitude or period.

It is any value that makes a denominator zero. These values must be stated even if a factor later cancels.

Cancelling a factor changes the appearance but not the original domain. Values that made the original denominator zero remain excluded.

Find a common denominator. Then rewrite each fraction, add or subtract numerators, and keep the denominator restrictions.

Multiply by the reciprocal of the second expression, factor when possible, cancel common factors, and record all non-permissible values from the original expression.

Multiplying by denominators can hide values that make an original denominator zero. Check proposed solutions against the original restrictions.

It usually comes from a denominator zero that does not cancel with the numerator. The function is undefined there and the graph approaches without crossing in the local model.

State restrictions, define variables, show formulas or transformations used, include units when applicable, and check that the result fits the context.