Intersection of Sets
The intersection of sets A and B, written A (upside down U) B, is the set of elements that are in both A and B.
Union of Sets
The union of sets A and B, written A U B, is the set of elements in either A or B (or in both).
Order of Operation in Evaluating Expressions
1. Do operations within parentheses or other grouping
2. Within grouping symbols, or if there are no grouping symbols:
a. Do all powers from left to right
b. Do all multiplications and divisions from left to right.
c. Do all additions and subtractions from left to right.
In a right triangle with legs of lengths a and b and hypotenuse of length c,
a2 + b2 = c2.
Area Model for Multiplication (discrete form)
The number of elements in a rectangular array with r rows and c columns is rc.
Multiplying Fractions Property
For all real numbers a, b, c, and d, with b and d not zero,
a/b • c/d = ac/bd.
Rate FActor Model for Multiplication
When a rate r is multiplied by another quantity x, the product is rx. So the unit of rx is the product of the units for r and x.
Rules for Multiplying Positive and Negative Numbers
If two numbers have the same sign, their product is positive. If two numbers have different signs, their product is negative.
Properties of Multiplication of Positive and Negative Numbers
1. The product of an odd number of negative numbers is negative.
2. The product of an even number of negative numbers is positive.
Multiplication Counting Principle
If one choice can be made in m ways and a second choice can be made in n ways, then there are mn ways of making the first choice followed by the second choice.
There are n! possible permutations of n different items when each item is used exactly once.
Putting-Together Model for Addition
If a quantity x is put together with a quantity y with the same units, and there is no overlap, then the result is the quantity x + y.
Distributive Property: Adding or Subtracting Like Terms
For any real numbers a, b, and c,
ac + bc = (a + b)c and
ac - bc = (a - b)c.
The Distributive Property: Removing Parentheses
For all real numbers a, b, and c,
c(a + b) = ca + cb and
c(a - b) = ca - cb.
Distributive Property: Adding Fractions
For all real numbers a, b, and c, with c ≠ 0, a/c + b/c = a + b/c.
Take-Away Model for Subtraction
If a quantity y is taken away from an original quantity x, the quantity left is x - y.
Comparison Model for Subtraction
The quantity x - y tells how much the quantity x differs from the quantity y.
Part 1: If A, B, and C are any three points, then AB + BC ≥ AC.
Part 2: If A, B, and C are vertices of a triangle, then AB + BC > AC.