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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

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.

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.

Pythagorean Theorem

In a right triangle with legs of lengths a and b and hypotenuse of length c,

a2 + b2 = c2.

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.

a/b • c/d = ac/bd.

Equal Fractions Property

If b ≠ 0 and k ≠ 0, then

ak/bk = a/b.

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.

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.

Permutation Theorem

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.

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.

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.

if a < b,

then a + c < b + c.

Algebraic Definition of Subtraction

For all real numbers a and b,

a - b = a + -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.

-(a + b) = -a + -b = -a - -b.

Opposite of a Difference Property

For all real number a and b,

-(a - b) = -a + b.

-(a - b) = -a + b.

Triangle Sum Theorem

If any triangle with angle measures a, b, and c in degrees,

a + b + c = 180.

a + b + c = 180.

Triangle Inequality

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.

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.

x - y < z < x + y.