Set of Ordered Pairs (x, y)
Consider the set notation {(x, y) | x, y ∈ ℤ, x + y = 2}. Here’s how to understand and verbalize each part:
- { }: The curly braces indicate the beginning and end of the set.
- (x, y): This represents an ordered pair of elements, where the first element is x and the second element is y.
- |: The vertical bar is read as “such that” and separates the description of the elements from the conditions they must satisfy.
- x, y ∈ ℤ: This is read as “x and y are elements of the set of integers.” The symbol ∈ means “is an element of,” and ℤ represents the set of all integers.
- x + y = 2: This is the condition that the elements must satisfy. In this case, the sum of x and y must be equal to 2.
Putting it all together, the set notation {(x, y) | x, y ∈ ℤ, x + y = 2} can be verbalized as “the set of all ordered pairs of integers (x, y) such that the sum of x and y is equal to 2.”
Some examples of members of this set include:
- (0, 2): Since 0 + 2 = 2, this ordered pair satisfies the condition.
- (1, 1): Since 1 + 1 = 2, this ordered pair also satisfies the condition.
- (-1, 3): Since -1 + 3 = 2, this ordered pair satisfies the condition as well.
These examples are included in the set because they are ordered pairs of integers that satisfy the given condition x + y = 2.
Set of Ordered Pairs (x, y)
Consider the set notation {(x, y) | x, y ∈ ℤ, 2x – y = 3}. Here’s how to understand and verbalize each part:
- { }: The curly braces indicate the beginning and end of the set.
- (x, y): This represents an ordered pair of elements, where the first element is x and the second element is y.
- |: The vertical bar is read as “such that” and separates the description of the elements from the conditions they must satisfy.
- x, y ∈ ℤ: This is read as “x and y are elements of the set of integers.” The symbol ∈ means “is an element of,” and ℤ represents the set of all integers.
- 2x – y = 3: This is the condition that the elements must satisfy. In this case, twice the value of x minus y must be equal to 3.
Putting it all together, the set notation {(x, y) | x, y ∈ ℤ, 2x – y = 3} can be verbalized as “the set of all ordered pairs of integers (x, y) such that twice the value of x minus y is equal to 3.”
Some examples of members of this set include:
- (0, -3): Since 2(0) – (-3) = 3, this ordered pair satisfies the condition.
- (2, 1): Since 2(2) – 1 = 3, this ordered pair also satisfies the condition.
- (3, 3): Since 2(3) – 3 = 3, this ordered pair satisfies the condition as well.
These examples are included in the set because they are ordered pairs of integers that satisfy the given condition 2x – y = 3.
Set of Ordered Pairs (x, y)
Consider the set notation {(x, y) | x, y ∈ ℝ, x² + y² = 4}. Here’s how to understand and verbalize each part:
- { }: The curly braces indicate the beginning and end of the set.
- (x, y): This represents an ordered pair of elements, where the first element is x and the second element is y.
- |: The vertical bar is read as “such that” and separates the description of the elements from the conditions they must satisfy.
- x, y ∈ ℝ: This is read as “x and y are elements of the set of real numbers.” The symbol ∈ means “is an element of,” and ℝ represents the set of all real numbers.
- x² + y² = 4: This is the condition that the elements must satisfy. In this case, the sum of the squares of x and y must be equal to 4.
Putting it all together, the set notation {(x, y) | x, y ∈ ℝ, x² + y² = 4} can be verbalized as “the set of all ordered pairs of real numbers (x, y) such that the sum of the squares of x and y is equal to 4.”
This equation represents a circle with radius 2 centered at the origin. Some examples of members of this set include:
- (2, 0): Since 2² + 0² = 4, this ordered pair satisfies the condition.
- (0, -2): Since 0² + (-2)² = 4, this ordered pair also satisfies the condition.
- (√2, √2): Since (√2)² + (√2)² = 4, this ordered pair satisfies the condition as well.
These examples are included in the set because they are ordered pairs of real numbers that satisfy the given condition x² + y² = 4.
Set of Ordered Pairs (x, y)
Consider the set notation {(x, y) | x, y ∈ ℝ, y = x² – 1}. This is another example of the Set of Ordered Pairs that we have been exploring. Here’s how to understand and verbalize each part:
- { }: The curly braces indicate the beginning and end of the set.
- (x, y): This represents an ordered pair of elements, where the first element is x and the second element is y.
- |: The vertical bar is read as “such that” and separates the description of the elements from the conditions they must satisfy.
- x, y ∈ ℝ: This is read as “x and y are elements of the set of real numbers.” The symbol ∈ means “is an element of,” and ℝ represents the set of all real numbers.
- y = x² – 1: This is the condition that the elements must satisfy. In this case, y must be equal to the square of x minus 1.
Putting it all together, the set notation {(x, y) | x, y ∈ ℝ, y = x² – 1} can be verbalized as “the Set of Ordered Pairs of real numbers (x, y) such that y is equal to the square of x minus 1.”
This equation represents a parabola. Some examples of members of this set include:
- (0, -1): Since 0² – 1 = -1, this ordered pair satisfies the condition.
- (1, 0): Since 1² – 1 = 0, this ordered pair also satisfies the condition.
- (-1, 0): Since (-1)² – 1 = 0, this ordered pair satisfies the condition as well.
These examples are included in the Set of Ordered Pairs because they are ordered pairs of real numbers that satisfy the given condition y = x² – 1.
Set of Ordered Pairs Satisfying y = sin(x)
Set-builder notation: {(x, y) | x, y ∈ ℝ, y = sin(x)}
Explanation:
- Domain and Range: The set includes all ordered pairs (x, y) where both x and y are real numbers. The domain includes all real numbers, and the range is restricted by the sine function, which has values between -1 and 1.
- Equation Constraint: The relationship between x and y is defined by the equation y = sin(x). This means that for any value of x, the corresponding value of y must be the sine of x.
- Graphical Representation: Graphically, this set represents the graph of the sine function, which is a wave-like pattern oscillating between -1 and 1. It has a period of 2π and an amplitude of 1.
- Examples of Members: Some examples of members of this set include:
- (0, 0) since sin(0) = 0
- (π/2, 1) since sin(π/2) = 1
- (π, 0) since sin(π) = 0
- (3π/2, -1) since sin(3π/2) = -1
- Applications: This set notation can be used to describe the behavior of oscillatory systems, such as sound waves, light waves, or mechanical vibrations.
Conclusion
The set {(x, y) | x, y ∈ ℝ, y = sin(x)} is a comprehensive way to describe the sine function. It includes all possible values of x and the corresponding values of y that satisfy the equation y = sin(x). This set notation provides a concise and precise way to represent this mathematical relationship, and it can be applied in various fields such as physics, engineering, and mathematics.
Ordered Pairs Representing the Cosine Function
Set-builder notation: {(x, y) | x, y ∈ ℝ, y = cos(x)}
Explanation:
- Domain and Range: The set includes all ordered pairs (x, y) where both x and y are real numbers. The domain includes all real numbers, and the range is restricted by the cosine function, which has values between -1 and 1.
- Equation Constraint: The relationship between x and y is defined by the equation y = cos(x). This means that for any value of x, the corresponding value of y must be the cosine of x.
- Graphical Representation: Graphically, this set represents the graph of the cosine function, which is also a wave-like pattern oscillating between -1 and 1. It has a period of 2π and an amplitude of 1.
- Examples of Members: Some examples of members of this set include:
- (0, 1) since cos(0) = 1
- (π/2, 0) since cos(π/2) = 0
- (π, -1) since cos(π) = -1
- (3π/2, 0) since cos(3π/2) = 0
- Applications: This set notation can be used to describe waveforms in various fields such as signal processing, acoustics, and electrical engineering.
Conclusion
The set {(x, y) | x, y ∈ ℝ, y = cos(x)} is a detailed way to describe the cosine function. It encompasses all possible values of x and the corresponding values of y that satisfy the equation y = cos(x). This set notation offers a clear and accurate way to represent this mathematical relationship, with applications in diverse scientific and engineering disciplines.