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 symbols. 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.
Square of the Square Root Property
For any nonnegative number n, √n • √n = n.
In a right triangle with legs of lengths a and b and hypotenuse of length c, a2 + b2 = c2.
Area Model for Multiplication
The area A of a rectangle with length L and width w is Ew.
Commutative Property of Multiplication
For any real numbers a and b, ab = ba.
Area Model for Multiplication (discrete form)
The number of elements in a rectangular array with r rows and c columns is rc.
Associative Property of Multiplication
For any real numbers a, b, and c, (ab)c = a(bc).
Multiplicative Identity Property of 1
For any real number a, a • 1 = 1 • a = a.
Property of Reciprocals
Suppose a ≠ 0. The reciprocal of a is 1/a. That is, a • 1/a = 1/a • a = 1.
Reciprocal of a Fraction Property
Suppose a ≠ 0 and b ≠ 0. The reciprocal of a/b is b/a.
Multiplication Property of Zero
For any real number a, a • 0 = 0 • a = 0.
Multiplying Fractions Property
For all real numbers a, b, c, and d, with b and d not zero, a/b • c/d = ac/bd.
Equal Fractions Property
If b ≠ 0 and k ≠ 0, then ak/bk = a/b.
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.
Multiplication Property of -1
For any real number a, a • -1 = -1 • a = -a.
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 Property of Equality
For all real numbers a, b, and c, if a = b, then ca = cb.
Multiplication Property of Inequality (Part 1)
If x < y and a is positive, then ax < ay.
Multiplication Property of Inequality (Part 2)
If x < y and a is negative, then ax > ay.
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.
Slide Model for Addition
If a slide x is followed by a slide y, the result is the slide x + y.
Commutative Property of Addition
For any real numbers a and b, a + b = b + a.
Associative Property of Addition
For any real number a, b, and c, (a + b) + c = a + (b + c).
Additive Identity Property
For any real number a, a + 0 = 0 + a = a.
Property of Opposites
For any real number a, a + -a = -a + a = 0.
Opposite of Opposites (Op-op) Property
For any real number a, -(-a) = a.
Addition Property of Equality
For all real numbers a, b, and c, if a = b, then a + c = b + c.
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.
Addition Property of Inequality
For all real numbers a, b, and c, if a < b, then a + c < b + c.
Algebraic Definition of Subtraction
For all real numbers a and b, a - b = a + -b.
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.
Opposite of a Sum Property
For all real numbers a and b, -(a + b) = -a + -b = -a - -b.
Opposite of a Difference Property
For all real number a and b, -(a - b) = -a + b.
Triangle Sum Theorem
If any triangle with angle measures a, b, and c in degrees, a + b + c = 180.
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.
Third Side Property
If x and y are the lengths of two sides of a triangle, and x > y, then the length z of the third side must satisfy the inequality x - y < z < x + y.