Which sets of ordered pairs represent functions and why?
{(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)}
{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}
{(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}
{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}
The sets of ordered pairs that represent functions are:
1. {(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)}:
This set is not a function because the input value 3 is associated with two different output values, 4 and 5.
2. {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}:
This set is a function because each input value is associated with a unique output value. In other words, there are no repeated input values.
3. {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}:
This set is a function because it represents a vertical line where each input value is associated with the same output value. While the output value is the same for all input values, it still fulfills the definition of a function.
4. {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}:
This set is not a function because the input value 1 is associated with multiple output values, 1, 2, 3, 4, and 5.
given the following function name the different ordered pair that would break this function (-18,12),(-4,2),(0,-8),(7,21).
To determine whether the ordered pairs (-18,12), (-4,2), (0,-8), and (7,21) break the function, we need to see if they follow the input-output relationship defined by the function. Without knowing the specific function, we cannot definitively say whether these ordered pairs break the function. The function could be any relationship between the input and output values, so it's not possible to determine solely based on the ordered pairs provided.
Objectives
Understand the meaning of a universal set.
Understand the basic ideas of a Venn diagram.
Use Venn diagrams to visualize relationships between two sets.
Find the complement of a set
Find the intersection of two sets.
Find the union of two sets.
Perform operations with sets.
Determine sets involving set operations from a Venn diagram.
Understand the meaning of and and or.
Use the formula for n (A U B) to solve some problem
A universal set is a set that contains all the possible elements or objects of interest in a particular context. It represents the entire set of items that are being considered or studied.
A Venn diagram is a graphical representation of sets using circles or other shapes. It is used to visually show the relationships and overlaps between different sets. Each set is represented by a circle, and the overlapping regions represent the elements that belong to both sets.
In a Venn diagram, the overlapping region between two sets represents the intersection or common elements between the sets. The non-overlapping regions represent the elements that are unique to each set.
The complement of a set refers to all the elements that do not belong to that set within the universal set. It is denoted as A' and contains all the elements that are present in the universal set but not in set A.
The intersection of two sets, denoted as A ∩ B, refers to the set of elements that are common to both sets. It contains only the elements that belong to both set A and set B.
The union of two sets, denoted as A U B, refers to the set of all elements that belong to either set A or set B or both. It contains all the elements from both sets without any repetition.
To perform operations with sets, we can use set notation and set operations. Set notation includes symbols like ∩ (intersection), U (union), and ' (complement), and set operations include finding the intersection, union, and complement of sets.
A Venn diagram can be used to represent sets and set operations visually. By drawing the circles or shapes that represent the sets and identifying the overlapping regions or non-overlapping regions, we can determine the elements involved in set operations.
The meaning of "and" and "or" in set operations is similar to their meaning in logic. In set operations, "and" represents the intersection of sets, where only the elements that belong to both sets are selected. "Or" represents the union of sets, where the elements that belong to either set or both are selected.
The formula for (A U B) represents the number of elements in the union of sets A and B. It can be calculated using the formula: n(A U B) = n(A) + n(B) - n(A ∩ B), where n represents the number of elements in a set. By substituting the values of n(A), n(B), and n(A ∩ B) into the formula, we can find the number of elements in the union of sets A and B.
given the universal set {1, 4 ,2, 6 , 9, 10,3,5,11, 14} that the element in set A={1, 4 ,2, 6 , 9, 10,3,5}: and that of set B={1,2,4,5,7,3,11,10,}. find the A U B , (A ∩ B)' and A ∩ B?
Using the given universal set {1, 4, 2, 6, 9, 10, 3, 5, 11, 14}, set A={1, 4, 2, 6, 9, 10, 3, 5}, and set B={1, 2, 4, 5, 7, 3, 11, 10}, we can find the following:
A U B (the union of sets A and B) represents all the elements that belong to either set A or set B or both. It contains all the unique elements from both sets with no repetition.
A U B = {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14}
(A ∩ B)' (the complement of the intersection of sets A and B) represents all the elements in the universal set that do not belong to the intersection of sets A and B.
First, let's find the intersection of sets A and B:
A ∩ B = {1, 2, 4, 3, 5, 10}
Next, let's find the complement of the intersection:
(A ∩ B)' = {6, 9, 7, 11, 14}
A ∩ B (the intersection of sets A and B) represents all the elements that are common to both sets. It contains only the elements that belong to both set A and set B.
A ∩ B = {1, 2, 4, 3, 5, 10}
To summarize,
A U B = {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14}
(A ∩ B)' = {6, 9, 7, 11, 14}
A ∩ B = {1, 2, 4, 3, 5, 10}
Given:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = { 1, 3, 7, 9 }
B = { 3, 7, 8, 10 }
Find ; A U B’ ,
A’ ∩ B and (A U B)’
To find A U B', we need to find the union of set A and the complement of set B.
Complement of set B (B') will consist of all the elements in the universal set U that are not in set B.
B' = {1, 2, 4, 5, 6, 7, 9}
A U B' will include all the elements that belong to set A or set B' or both.
A U B' = {1, 3, 7, 9, 2, 4, 5, 6}
To find A' ∩ B, we need to find the intersection of the complement of set A and set B.
Complement of set A (A') will consist of all the elements in the universal set U that are not in set A.
A' = {2, 4, 5, 6, 8, 10}
A' ∩ B will include all the elements that belong to set A' and set B.
A' ∩ B = {3, 7}
To find (A U B)', we need to find the complement of the union of set A and set B.
A U B will include all the elements that belong to set A or set B or both.
A U B = {1, 3, 7, 8, 10}
The complement of the union (A U B)' will consist of all the elements in the universal set U that are not in set A U B.
(A U B)' = {2, 4, 5, 6, 9}
To summarize:
A U B' = {1, 3, 7, 9, 2, 4, 5, 6}
A' ∩ B = {3, 7}
(A U B)' = {2, 4, 5, 6, 9}
To determine whether a set of ordered pairs represents a function, we need to check if each input (x-value) has a unique output (y-value). In other words, there should not be two or more ordered pairs with the same x-value but different y-values.
Let's examine each set of ordered pairs:
1. {(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)}
In this set, we can see that there are two ordered pairs with the same x-value of 3 but different y-values (4 and 5). Thus, this set is NOT a function.
2. {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}
This set has a unique x-value for each ordered pair, and each x-value has a corresponding unique y-value. Therefore, this set represents a function.
3. {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}
In this set, all the ordered pairs have the same y-value (1). Even though each x-value has a unique output, the fact that all outputs are the same means that this set represents a function. Specifically, it represents a horizontal line.
4. {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}
In this set, all the ordered pairs have the same x-value (1) but different y-values. This violates the rule of a function, which requires each x-value to have a unique output. Therefore, this set is NOT a function.
In summary:
- Set 2 {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} represents a function.
- Sets 1 {(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)}, 3 {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}, and 4 {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)} do not represent functions.