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Questions and Answers
The equation (A∩B)∪(A – B) = A is proven using properties like the ______ property.
The equation (A∩B)∪(A – B) = A is proven using properties like the ______ property.
distributive
The set of ______ numbers, denoted by N, includes the numbers 1, 2, 3, and so on.
The set of ______ numbers, denoted by N, includes the numbers 1, 2, 3, and so on.
natural
The set of ______, denoted by Z, includes all positive and negative whole numbers, as well as zero.
The set of ______, denoted by Z, includes all positive and negative whole numbers, as well as zero.
integers
The set of positive integers, denoted by Z+, is equivalent to the set of ______ numbers.
The set of positive integers, denoted by Z+, is equivalent to the set of ______ numbers.
The set of ______ numbers, denoted by Q, includes numbers that can be expressed as a fraction a/b, where a and b are integers and b ≠ 0.
The set of ______ numbers, denoted by Q, includes numbers that can be expressed as a fraction a/b, where a and b are integers and b ≠ 0.
A decimal representation of a rational number is either terminating or ______.
A decimal representation of a rational number is either terminating or ______.
To express the repeating decimal 4.83 as a rational number, you can set x = 4.83 and then find expressions for 10x and ______ to eliminate the repeating part.
To express the repeating decimal 4.83 as a rational number, you can set x = 4.83 and then find expressions for 10x and ______ to eliminate the repeating part.
In the proof (A∩B)∪(A – B) = A ∩(A∪Bʹ), the step (A∪Bʹ)∩U = A∪Bʹ is justified because any set intersected with the ______ set remains unchanged.
In the proof (A∩B)∪(A – B) = A ∩(A∪Bʹ), the step (A∪Bʹ)∩U = A∪Bʹ is justified because any set intersected with the ______ set remains unchanged.
The set difference A – B is defined as the set of all elements $x$ such that $x$ is in A and $x$ is not in ______.
The set difference A – B is defined as the set of all elements $x$ such that $x$ is in A and $x$ is not in ______.
If U is the universal set, the ______ of a set A, denoted by A', contains all elements in U that are not in A.
If U is the universal set, the ______ of a set A, denoted by A', contains all elements in U that are not in A.
A ______ diagram is a pictorial representation of sets, often using circles within a rectangle to represent the universal set.
A ______ diagram is a pictorial representation of sets, often using circles within a rectangle to represent the universal set.
The ______ of two sets A and B, denoted A ∩ B, includes all elements that are present in both A and B.
The ______ of two sets A and B, denoted A ∩ B, includes all elements that are present in both A and B.
Two sets A and B are considered ______ if their intersection results in an empty set (A ∩ B = ∅).
Two sets A and B are considered ______ if their intersection results in an empty set (A ∩ B = ∅).
The ______ of two sets A and B, denoted A ∪ B, encompasses all elements found in either A or B, or both.
The ______ of two sets A and B, denoted A ∪ B, encompasses all elements found in either A or B, or both.
According to De Morgan's Laws, the complement of the union of two sets (A ∪ B)' is equal to the intersection of their individual ______ A' ∩ B'.
According to De Morgan's Laws, the complement of the union of two sets (A ∪ B)' is equal to the intersection of their individual ______ A' ∩ B'.
If set B is a subset of set A (B ⊂ A), it logically follows that the complement of A (A') is a subset of the ______ of B (B').
If set B is a subset of set A (B ⊂ A), it logically follows that the complement of A (A') is a subset of the ______ of B (B').
The proof that $\sqrt{2}$ is not a rational number relies on making an ______ about $\sqrt{2}$ and showing that this leads to contradictory statements.
The proof that $\sqrt{2}$ is not a rational number relies on making an ______ about $\sqrt{2}$ and showing that this leads to contradictory statements.
If $m^2$ has a factor of 2, then $m^2$ is ______, which implies that $m$ itself must be divisible by 2.
If $m^2$ has a factor of 2, then $m^2$ is ______, which implies that $m$ itself must be divisible by 2.
To show that $\sqrt{2} + \sqrt{3}$ is not a rational number, one begins by supposing that $\sqrt{2} + \sqrt{3}$ is ______ and manipulates the equation to arrive at a contradiction.
To show that $\sqrt{2} + \sqrt{3}$ is not a rational number, one begins by supposing that $\sqrt{2} + \sqrt{3}$ is ______ and manipulates the equation to arrive at a contradiction.
A ______ number is defined as one that can be expressed as a quotient of two integers.
A ______ number is defined as one that can be expressed as a quotient of two integers.
An irrational number, such as $\pi$ or $\sqrt{2}$, cannot be expressed as a simple ______ of two integers.
An irrational number, such as $\pi$ or $\sqrt{2}$, cannot be expressed as a simple ______ of two integers.
The union of the the set of ______ numbers with the set of irrational numbers creates the set of real numbers, denoted as R.
The union of the the set of ______ numbers with the set of irrational numbers creates the set of real numbers, denoted as R.
The set of real numbers contains the set of natural numbers and the set of ______.
The set of real numbers contains the set of natural numbers and the set of ______.
When representing intervals on the real number line, a ______ bracket, such as ( or ), indicates that the number at the boundary is not included in the interval.
When representing intervals on the real number line, a ______ bracket, such as ( or ), indicates that the number at the boundary is not included in the interval.
To eliminate the decimal part when converting a repeating decimal to a fraction, subtracting 100x minus 10x from $435.87$ results in 90x = ______.
To eliminate the decimal part when converting a repeating decimal to a fraction, subtracting 100x minus 10x from $435.87$ results in 90x = ______.
When converting $0.354$ (with the 354 repeating) to a fraction, setting $x = 0.354$ implies $1000x = 354.354$, and subtracting $1000x - x$ isolates the repeating part, giving 999x = ______.
When converting $0.354$ (with the 354 repeating) to a fraction, setting $x = 0.354$ implies $1000x = 354.354$, and subtracting $1000x - x$ isolates the repeating part, giving 999x = ______.
To demonstrate that $2.75$ is a rational number, multiplying by a suitable power of 10 removes the decimal, so $100x = 275$, thus $x = 275/100$ which simplifies to ______.
To demonstrate that $2.75$ is a rational number, multiplying by a suitable power of 10 removes the decimal, so $100x = 275$, thus $x = 275/100$ which simplifies to ______.
An irrational number is defined as a number that cannot be expressed in the form a/b for any integers a and b, and its decimal representation is non-terminating and ______.
An irrational number is defined as a number that cannot be expressed in the form a/b for any integers a and b, and its decimal representation is non-terminating and ______.
If $p^2$ is divisible by 2, then the proof by contradiction demonstrates that $p$ itself must be ______ by 2.
If $p^2$ is divisible by 2, then the proof by contradiction demonstrates that $p$ itself must be ______ by 2.
In a proof by ______, one assumes the opposite of what needs to be proven. This assumption is then followed to a logical, yet contradictory, conclusion, thereby affirming the original statement.
In a proof by ______, one assumes the opposite of what needs to be proven. This assumption is then followed to a logical, yet contradictory, conclusion, thereby affirming the original statement.
To prove that $\sqrt{2}$ is not a rational number, one starts by assuming that it ______ can be expressed as a fraction a/b, where a and b are integers.
To prove that $\sqrt{2}$ is not a rational number, one starts by assuming that it ______ can be expressed as a fraction a/b, where a and b are integers.
When expressing a repeating decimal as a fraction, the goal is to eliminate the repeating part through a process that often involves ______ and subtraction, hence simplifying the number into a rational form.
When expressing a repeating decimal as a fraction, the goal is to eliminate the repeating part through a process that often involves ______ and subtraction, hence simplifying the number into a rational form.
The ________ law states that for any real numbers a, b, the sum a + b is always equal to b + a.
The ________ law states that for any real numbers a, b, the sum a + b is always equal to b + a.
According to the ________ law of addition, the way in which numbers are grouped does not change their sum: a + (b + c) = (a + b) + c.
According to the ________ law of addition, the way in which numbers are grouped does not change their sum: a + (b + c) = (a + b) + c.
The ________ identity states that for any real number 'a', a + 0 = a. Here, adding zero does not change the original value.
The ________ identity states that for any real number 'a', a + 0 = a. Here, adding zero does not change the original value.
The ________ identity property is demonstrated when any real number 'a' is multiplied by 1, resulting in 'a': a * 1 = a.
The ________ identity property is demonstrated when any real number 'a' is multiplied by 1, resulting in 'a': a * 1 = a.
Every real number 'a' has an ________ inverse '-a' such that a + (-a) = 0, illustrating the concept of balancing values to reach zero.
Every real number 'a' has an ________ inverse '-a' such that a + (-a) = 0, illustrating the concept of balancing values to reach zero.
The ________ law, a(b + c) = ab + ac, shows how to distribute a single term across multiple terms inside parentheses, simplifying expressions.
The ________ law, a(b + c) = ab + ac, shows how to distribute a single term across multiple terms inside parentheses, simplifying expressions.
The ________ property posits that for any two real numbers x and y, only one of the following is true: x > y, x < y, or x = y.
The ________ property posits that for any two real numbers x and y, only one of the following is true: x > y, x < y, or x = y.
In solving the linear equation $ax + b = c$, by adding $(-b)$ to both sides we get $ax = c - b$. This step uses the property of the ________ inverse.
In solving the linear equation $ax + b = c$, by adding $(-b)$ to both sides we get $ax = c - b$. This step uses the property of the ________ inverse.
When adding complex numbers $(3 + 5i)$ and $(2 - 3i)$, we add the real and ______ parts separately to get $5 + 2i$.
When adding complex numbers $(3 + 5i)$ and $(2 - 3i)$, we add the real and ______ parts separately to get $5 + 2i$.
To simplify $(4 + i)^2$, we expand it as $(4 + i)(4 + i)$, which equals $16 + 8i + i^2$. Then, knowing that $i^2 = -1$, we find that $(4 + i)^2$ simplifies to ______ + 8i.
To simplify $(4 + i)^2$, we expand it as $(4 + i)(4 + i)$, which equals $16 + 8i + i^2$. Then, knowing that $i^2 = -1$, we find that $(4 + i)^2$ simplifies to ______ + 8i.
When dividing complex numbers, for example $\frac{2 - 5i}{3 + 2i}$, we multiply both the numerator and the denominator by the ______ of the denominator.
When dividing complex numbers, for example $\frac{2 - 5i}{3 + 2i}$, we multiply both the numerator and the denominator by the ______ of the denominator.
If $z = \frac{3 + 2i}{4 - 3i}$, to express $z$ in the form $a + ib$, we multiply the numerator and denominator by the conjugate of the denominator, which simplifies to $z = \frac{6}{25} + \frac{17}{25}i$. Therefore, the real part, $a$, of $z$ is ______.
If $z = \frac{3 + 2i}{4 - 3i}$, to express $z$ in the form $a + ib$, we multiply the numerator and denominator by the conjugate of the denominator, which simplifies to $z = \frac{6}{25} + \frac{17}{25}i$. Therefore, the real part, $a$, of $z$ is ______.
Given $z = \frac{3 + 2i}{4 - 3i}$, after expressing it in the form $a + bi$ and finding $a$ and $b$, the magnitude squared of $z$, denoted as $|z|^2$, is calculated as $a^2 + b^2$. Thus, $|z|^2$ equals ______.
Given $z = \frac{3 + 2i}{4 - 3i}$, after expressing it in the form $a + bi$ and finding $a$ and $b$, the magnitude squared of $z$, denoted as $|z|^2$, is calculated as $a^2 + b^2$. Thus, $|z|^2$ equals ______.
To solve for $x$ and $y$ in the equation $\frac{x + iy}{2 + i} = 3 - 5i$, we cross-multiply to get $x + iy = (3 - 5i)(2 + i)$. After expanding and simplifying, we equate the real and imaginary parts to find that $x$ equals ______.
To solve for $x$ and $y$ in the equation $\frac{x + iy}{2 + i} = 3 - 5i$, we cross-multiply to get $x + iy = (3 - 5i)(2 + i)$. After expanding and simplifying, we equate the real and imaginary parts to find that $x$ equals ______.
Given the equation $\frac{1}{x + iy} = \frac{3 - 2i}{1 + i}$, to find $x$ and $y$, we cross-multiply to obtain $1 + i = (3 - 2i)(x + iy)$. After expanding the right side and equating real and imaginary parts, we can solve for $x$ and $y$, assuming that $x + iy$ is not equal to ______
Given the equation $\frac{1}{x + iy} = \frac{3 - 2i}{1 + i}$, to find $x$ and $y$, we cross-multiply to obtain $1 + i = (3 - 2i)(x + iy)$. After expanding the right side and equating real and imaginary parts, we can solve for $x$ and $y$, assuming that $x + iy$ is not equal to ______
In the expression $\frac{2 - 5i}{3 + 2i}$, multiplying the numerator and denominator by $3 - 2i$ is a strategic step because it eliminates the ______ part from the denominator.
In the expression $\frac{2 - 5i}{3 + 2i}$, multiplying the numerator and denominator by $3 - 2i$ is a strategic step because it eliminates the ______ part from the denominator.
Flashcards
Set Difference
Set Difference
A – B = {x ⏐ x ∈ A and x ∉ B} shows elements in A but not in B.
Complement of a Set
Complement of a Set
Aʹ = U – A contains all elements in the universal set not in A.
Venn Diagram
Venn Diagram
A pictorial representation to show relationships between sets, like subsets and overlaps.
Set Intersection
Set Intersection
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Disjoint Sets
Disjoint Sets
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Set Union
Set Union
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Properties of Set Operations
Properties of Set Operations
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De Morgan's Laws
De Morgan's Laws
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Rational Numbers
Rational Numbers
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Decimal Representation of Rational Numbers
Decimal Representation of Rational Numbers
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Terminating Decimal
Terminating Decimal
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Repeating Decimal
Repeating Decimal
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Set of Natural Numbers (N)
Set of Natural Numbers (N)
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Set of Integers (Z)
Set of Integers (Z)
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Set of Real Numbers (R)
Set of Real Numbers (R)
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Set of Positive Integers (Z+)
Set of Positive Integers (Z+)
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Irrational Number
Irrational Number
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Non-Terminating Decimal
Non-Terminating Decimal
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Proof by Contradiction
Proof by Contradiction
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Power of 10 Multiplication
Power of 10 Multiplication
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2.75 as a Fraction
2.75 as a Fraction
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Square of an Integer
Square of an Integer
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Commutative Law
Commutative Law
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Associative Law
Associative Law
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Additive Identity
Additive Identity
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Multiplicative Identity
Multiplicative Identity
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Additive Inverse
Additive Inverse
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Multiplicative Inverse
Multiplicative Inverse
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Distributive Law
Distributive Law
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Trichotomy Law
Trichotomy Law
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Even Number
Even Number
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Set of Real Numbers
Set of Real Numbers
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Continuous Line
Continuous Line
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Intervals
Intervals
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Contradiction Example
Contradiction Example
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Adding Complex Numbers
Adding Complex Numbers
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Complex Multiplication
Complex Multiplication
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i Squared
i Squared
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Finding the Modulus
Finding the Modulus
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Conjugate of a Complex Number
Conjugate of a Complex Number
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Cross-Multiplication with Complex Numbers
Cross-Multiplication with Complex Numbers
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Real and Imaginary Parts
Real and Imaginary Parts
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Equating Complex Numbers
Equating Complex Numbers
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Study Notes
Sets Operations
- A set is a collection of well-defined objects called elements. Elements within a set must be distinct and distinguishable.
- Capital letters are used for set names and lowercase letters for set elements.
- Braces {} are used to represent sets; elements are listed between the braces, e.g., A = {1, 2, 3, 4}.
- Sets can also be described by defining a property or attribute, e.g., A = {x : x is a natural number}.
- x ∈ A means 'x belongs to A'.
- x ∉ A means 'x does not belong to A'.
Set Notation
- A = {2, 3, 4, 5, 6} , A = {n : n is a natural number and 1 < n < 7}
- A = {2, 4, 6, 8, 10} 4 ∈ A, 5 ∉ A.
- S = {1, 2, 3, 4, 10} and B = {k ∈ S : k = 3n + 1, n = 0, 1, 2, 3}, B = {1, 4, 7, 10}.
- A ⊆ B or B ⊃ A means 'A is a subset of B'. Every element in A is also in B.
- A = B, if both A ⊆ B and B ⊆ A.
- Ø denotes the empty set (null set) i.e., a set with no elements. The empty set is a subset of every set.
Set Operations
- Intersection (A∩B) is the set containing common elements of sets A and B. A∩B = {x | x ∈ A and x ∈ B}.
- Union (A∪B) is the set of all elements that belong to either set A or set B or both. A∪B = {x | x∈ A or x ∈ B}.
- Disjoint sets: A and B are disjoint if their intersection is the empty set (A∩B = Ø).
- Difference (A − B) contains elements in A but not in B; A − B = {x | x ∈ A and x ∉ B}.
- Complement (A'): The complement of A, written A', is the set of all elements in the universal set U that are not in A; A' = U – A.
- Venn diagram: A visual representation of sets.
- Commutativity: A∪B = B∪A; A∩B = B∩A.
- Associativity: AU(B∪C) = (A∪B)∪C; A∩(B∩C) = (A∩B)∩C.
Distributive properties
- A ∩ (B∪C) = (A∩B) ∪ (A∩C)
- A ∪ (B∩C) = (A∪B) ∩ (A∪C)
De Morgan's Laws
- (A ∪ B)' = A' ∩ B'
- (A ∩ B)' = A' ∪ B'
Sets of Numbers
- Natural numbers (N): {1, 2, 3, ...}
- Integers (Z): {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Rational numbers (Q): numbers that can be expressed as a fraction a/b, where a and b are integers and b ≠ 0.
- Irrational numbers (Irr): numbers that cannot be expressed as a fraction of integers.
- Real numbers (R): the union of rational and irrational numbers.
- Intervals: a subset of real numbers
- The real numbers R form a continuous line.
- Intervals can be represented using brackets: open brackets ( ) for excluding endpoints, closed brackets [ ] for including endpoints.
Properties of Real Numbers
- Commutative law of addition: a + b = b + a
- Commutative law of multiplication: ab = ba
- Associative law of addition: a + (b + c) = (a + b) + c
- Associative law of multiplication: a(bc) = (ab)c
- Additive identity: a + 0 = a
- Multiplicative identity: a × 1 = a
Order Relations
- x > y if x − y > 0 (x is greater than y)
- x < y if x − y < 0 (x is less than y)
- x = y if x − y = 0 (x is equal to y)
Absolute Value
- |x| = x if x ≥ 0
- |-x| = x if x < 0
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Description
Questions cover set theory, including operations like union, intersection, and set difference. It also covers number systems such as natural, integer, rational, and real numbers, along with their properties and representations.
