MATH 0309 Chapter 1&2 vocabulary/concepts

The expression a − b is equal to a + .

-b

If a = 0, and b ≠ 0, then a/b = _______.

0

If either a or b is zero then the product ab = _______.

0

The product of a number and its reciprocal is _______. For example -2/3⋅( )=1

1

If a is a nonzero real number, then the reciprocal of a is _______.

1/a

Given the expression x, the value of the coefficient is _______, and the exponent is _______.

1; 1

The fraction 4/4 can also be written as the whole number----------- , and the fraction 4/1 can be written as the whole number-------------- .

1; 4

The statement 5 ≠ 6 is read as "________."

5 is not equal to 6

The expression _______ is read as "8-squared."

8^2

What is a universal set?

A Universal Set is the set of all elements under consideration, denoted by capital U. All other sets are subsets of the universal set.

Explain the difference between a finite and an infinite set.

A set is called finite if it has no elements, or has cardinality that is a natural number. A set that is not finite is called an infinite set.

Explain what a set is

A set is well-defined if for any given object

Define the empty set and give two examples of an empty set.

A set with no elements is called an empty set or null set. The symbols used to represent the empty set are { } or Ø.

Explain the difference between a simple and a compound statement.

A simple statement contains only one idea. A compound statement is a statement formed by joining two or more simple statements with a connective.

Define the term statement in your own words.

A statement is a declarative sentence that can be objectively determined to be either true or false, but not both.

Explain why the empty set is a subset, but not a proper subset, of itself.

An empty set can be part of another set, but is

Explain why the negation of "All spring breaks are fun" is not "All spring breaks are not fun."

Because if some spring breaks are fun or some or not fun, that would already be a negation. It is not all or nothing.

Explain the difference between a subset and a proper subset.

If a set A is a subset of a set B and is not equal to B, then we call A a proper subset of B, and write A ⊂ B. That is, A ⊆ B and A ≠ B.

When are two sets said to be disjoint?

In mathematics, two sets are said to be disjoint sets if they have no element in common.

Explain why we're interested in writing statements in symbols.

Our main goal in the study of formal logic is to be able to evaluate logical arguments objectively. In order to do that, we'll need to write statements in symbolic form.

List and describe three ways to write sets.

Roster method - {2, 5, 7} Set Notation - 27 ∈ {1, 5, 9, 13, 17, . . .} or z ∉ {v, w, x, y, z} Descriptive Method - short verbal statement to describe the set Set Builder Notation - set {1, 2, 3, 4, 5, 6} can be written in set-builder notation as {x|x ∈ N and x < 7} Ellipsis - use an ellipsis to represent the missing elements as long as we illustrate a clear pattern. For example, the set {1, 2, 3, . . ., 99, 100} includes all the natural numbers from 1 to 100. The cardinal number of a set is the number of elements in the set. For a set A the symbol for the cardinality is n(A), which is read as "n of A."

What is the complement of a set?

The complement of a set A, symbolized A′, is the set of elements in the universal set that are not in A. Using set-builder notation, the complement of A is A′ = {x|x ∈ U and x ∉ A} .

Explain the difference between the union and intersection of two sets.

The intersection of two sets A and B, symbolized by A ∩ B, is the set of all elements that are in both sets. In set-builder notation, A ∩ B = {x|x ∈ A and x ∈ B}. The union of two sets A and B, symbolized by A ∪ B, is the set of all elements that are in either set A or set B (or both). In set-builder notation, A ∪ B = {x|x ∈ A or x ∈ B}

Write an example from real life that represents the difference of sets and explain why it represents difference.

The relative complement or set difference of sets A and B, denoted A - B, is the set of all elements in A that are not in B. In set-builder notation, A - B = {x ∈ U : x ∈ A and x ∉ B}= A ∩ B'. Example: For the lead-in example on the previous page, let the universal set U be the set of all U.S. dollars, let set A be the set of $836 Sam originally has in the checking account, and let B be the set of the $429 of the check. Then the set difference of A and B would be the $407 remaining in the checking account. Example: Let A = {a, b, c, d} and B = {b, d, e}. Then A - B = {a, c} and B - A = {e}. Example: Let G = {t, a, n} and H = {n, a, t}. Then G - H = ∅. Solution is the number (element) that is not present in both sets at the same time.

Write an example of a set that is well-defined, and one that is not. (No stealing examples from the book!)

The set "letters of the English alphabet" is well-defined since it consists of the 26 symbols we use to make up our alphabet, and no other objects. The set "tall people in your class" is not well-defined because who exactly belongs to that set is open to interpretation.

Describe the terms and symbols used for the four connectives.

There are four basic connectives used in logic: and (the conjunction), or (disjunction), if . . . then (conditional), and if and only if (biconditional). Conjunction ∧ (and) Disjunction ∨ (or) Conditional → (if then) Biconditional ↔ (if and only if)

What is the difference between equal and equivalent sets?

Two sets A and B are equal (written A = B) if they have exactly the same members or elements. Two finite sets A and B are said to be equivalent (written A ≅ B) if they have the same number of elements: that is, n(A) = n(B).

What is meant by "one-to-one correspondence between two sets"?

Two sets have a one-to-one correspondence of elements if each element in the first set can be paired with exactly one element of the second set and each element of the second set can be paired with exactly one element of the first set.

Write an example from real life that represents the union of sets and explain why it represents union. Then do the same for intersection.

Union of two sets. The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. In symbols, . For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6} then A ∪ B = {1, 2, 3, 4, 5, 6, 7}. Solution is all the numbers in both sets.

Is the sentence "This sentence is a statement" a statement? Explain.

Yes. It can be easily verified that it is.

What is a subset?

a grouping of the members within a set based on a shared characteristic. If every element of a set A is also an element of a set B, then A is called a subset of B. The symbol ⊆ is used to designate a subset; in this case, we write A ⊆ B.

The statement a < b is read as "________."

a is less than b

Which of the following expressions represents the product of 2 and x? a. 2x b. 2⋅x c. 2(x) d. (2)x e. (2)(x)

all of these

If a and b have different signs, and if ∣b∣ > ∣a∣, then the sum will have the same sign as (choose one: a or b).

b

In the expression bn, the value b is called the _______ and n is called the _______ or _______.

base; exponent; power

The statement c ≥ d is read as "_______________."

c is greater than or equal to d

The constant factor in a term is called the _______ of the term.

coefficient

Given the expression 12y + ab − 2x + 18, the terms 12y, ab, and −2x are variable terms, whereas 18 is a _______ term.

constant

Values that do not vary are called ________.

constants

The numbers being multiplied in a product are called----- .

factors

The statements a < b, a > b, and a ≠ b are examples of ________.

inequalities

The -------------common denominator of two or more fractions is the LCM of their denominators.

least

Terms that have the same variables, with corresponding variables raised to the same powers, are called _______ terms.

like

A fraction is said to be in --------------- terms if the numerator and denominator share no common factor other than 1.

lowest

The least common multiple (LCM) of two numbers is the smallest whole number that is a------------ of both numbers.

multiple

If a and b are both negative, then a + b will be (choose one: positive or negative).

negative

If a and b have different signs, then the quotient a/b is (choose one: positive or negative).

negative

Given a fraction a/b with b ≠ 0, the value a is the ------ and ------------ is the denominator.

numerator

Two numbers that are the same distance from 0 but on opposite sides of 0 on the number line are called ________.

opposites

The set of rules that tell us the order in which to perform operations to simplify an algebraic expression is called the _________________.

order or operations: P E M&D A&S

The expression _______ is read as "p to the 4th power."

p^4

If a and b have the same sign, then the product ab is (choose one: positive or negative).

positive

If a is positive and b is negative, then the difference a − b will be (choose one: positive or negative).

positive

A -------- is the result of multiplying two or more numbers.

product

Fill in the blanks with the words sum, difference, product, or quotient. The _______ of 10 and 2 is 5. The _______ of 10 and 2 is 20. The _______ of 10 and 2 is 12. The _______ of 10 and 2 is 8.

quotient; product; sum; difference

The symbol √ is called a _______ sign and is used to find the principal _______ root of a nonnegative real number.

radical; square

Two nonzero numbers a/b and b/a are ------------ because their product is 1.

reciprocals

In mathematics, a well-defined collection of elements is called a ________.

set

What is the negation of a statement?

the opposite of the original statement The negation of a statement is a corresponding statement with the opposite truth value. This means that if a statement is true its negation is false, and if a statement is false its negation is true.

What does it mean for a set to be well-defined?

to be well-defined, the definition of what is or is not in a set has to be based on facts, not opinions.

If a ≠ and b = 0, then a/b is _______.

undefined

A ________ is a symbol or letter used to represent an unknown number.

variable

The absolute value of a real number, a, is denoted by ________ and is the distance between a and ________ on the number line.

|a|; 0