1.1 Real Numbers and Sets and Intervals - College Algebra (Math 1150)

A set of practice flashcards covering key concepts from 1.1 Real Numbers, sets and intervals, absolute value, distance, and basic algebraic properties.

What set does N = {1, 2, 3, …} represent?

Natural Numbers

What set denotes Whole Numbers?

W = {0, 1, 2, 3, …}

What is the standard notation for Integers and what elements does it include?

Z = { …, -3, -2, -1, 0, 1, 2, 3, … }

How are Rational Numbers defined?

Q = { p/q : p ∈ Z, q ∈ N, q ≠ 0 }

How can rational numbers be written as decimals?

Decimals that terminate or repeat (periodic)

Give examples of rational numbers.

1/2, 1, 20

What is an irrational number?

A number that cannot be expressed as a ratio of two integers; decimals are non-terminating and non-repeating (e.g., √2, π)

What is the set of Real Numbers?

R = set of all rational and irrational numbers

What are two key decimal properties of irrational numbers?

Non-terminating and non-repeating decimals

On the real number line, where is the origin located?

The point corresponding to 0 is the Origin

Where are positive real numbers located on the real number line relative to the origin?

To the right of the origin

Where are negative real numbers located on the real number line relative to the origin?

To the left of the origin

True or False: -1.5 is a rational number?

True

True or False: 4 is an integer, a rational number, and a real number?

True

True or False: 81 is an irrational number?

False

True or False: -√12 is an irrational number?

True

What does A ∪ B denote in set notation?

The union of A and B (all elements in A or B or both, without repetition)

What does A ∩ B denote in set notation?

The intersection of A and B (elements common to both)

Given A = {−2, −1, 0, 1, 2}, B = {−4, −2, 0, 2, 4}, what is A ∩ B?

{−2, 0, 2}

Given A = {−2, −1, 0, 1, 2}, B = {−4, −2, 0, 2, 4}, what is A ∪ B?

{−4, −2, −1, 0, 1, 2, 4}

Given A = {−2, −1, 0, 1, 2} and C = {−3, 3}, what is A ∩ C?

∅ (empty set)

What is an open interval?

(a, b) with a < x < b; endpoints not included

What is a closed interval?

[a, b] with a ≤ x ≤ b; endpoints included

What is an unbounded interval?

An interval extending indefinitely, such as (a, ∞), (-∞, b), or (-∞, ∞)

How is interval notation related to set notation?

(a,b) = {x | a < x < b}; [a,b] = {x | a ≤ x ≤ b}

What is IUJ for I = (-3,4) and J = [2,6]?

(-3,6]

What is INJ for I = (-3,4) and J = [2,6]?

[2,4)

What is the distance between two real numbers a and b on the number line?

d(a,b) = |a − b|

What is the distance from -3 to 4?

7

What is the absolute value |a| for a real number a?

|a| = a if a ≥ 0; |a| = -a if a < 0

What is the geometric interpretation of |a| on the real line?

The distance of the point a from the origin

What does PEMDAS stand for?

Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)

Which operations are performed first inside grouping symbols according to PEMDAS?

Innermost grouping symbols first

What is the distributive property?

a(b+c) = ab + ac

What is the additive identity property?

There exists a unique real number 0 such that a + 0 = a

What is the multiplicative identity property?

There exists a unique real number 1 such that a · 1 = a

What is the additive inverse (opposite) property?

Every real number a has an additive inverse -a such that a + (-a) = 0

What is the multiplicative inverse property for nonzero a?

a · (1/a) = 1