Suppose U = {1, 2, 3, 4, 5, 6, 7, 8}, A = (1, 3, 5, 7}, and B = {4, 5, 6}. Tell whether each statement is true or false. A B
The statements are not provided. Could you please provide the statements for me to determine whether they are true or false?
A half circle with line under B
Given that U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 3, 5, 7}, and B = {4, 5, 6}, let's determine whether each statement is true or false.
1. A ⊆ U (A is a subset of U)
True, because all the elements of A are also elements of U.
2. B ⊆ U (B is a subset of U)
True, because all the elements of B are also elements of U.
3. A ∩ B = {} (Intersection of A and B is an empty set)
False, because the intersection of A and B is {5}, which is not an empty set.
4. A ∪ B = {1, 3, 4, 5, 6, 7} (Union of A and B)
False, because the union of A and B is {1, 3, 4, 5, 6, 7}, which also includes the element 4 that is not in A.
Therefore, the statements are as follows:
1. True
2. True
3. False
4. False
So what is the final answer
The final answer is as follows:
1. A ⊆ U (A is a subset of U) - True
2. B ⊆ U (B is a subset of U) - True
3. A ∩ B = {} (Intersection of A and B is an empty set) - False
4. A ∪ B = {1, 3, 4, 5, 6, 7} (Union of A and B) - False
To determine whether each statement is true or false, we need to compare the elements of sets A and B.
1. A ∩ B = {5} (Intersection of A and B): True
Explanation: The only element common between set A and set B is 5.
2. A ∪ B = {1, 3, 4, 5, 6, 7} (Union of A and B): True
Explanation: The union of set A and set B contains all the distinct elements present in both sets, without repetition.
3. A ⊆ U (Subset of U): True
Explanation: Set A is a subset of set U if all the elements of A are also present in U. In this case, all elements of A occur in set U, so A is a subset of U.
4. B ⊆ A (Subset of A): False
Explanation: Set B is a subset of set A if all the elements of B are also present in A. However, 4 is not present in set A, so B is not a subset of A.
To determine whether each statement is true or false, we need to compare the elements of set A with set B.
Statement 1: A ∪ B
The symbol ∪ represents the union of sets, which is a set that contains all the distinct elements from both sets A and B. To find A ∪ B, we need to combine the elements from both sets:
A = {1, 3, 5, 7}
B = {4, 5, 6}
A ∪ B = {1, 3, 4, 5, 6, 7}
Statement 2: A ∩ B
The symbol ∩ represents the intersection of sets, which is a set that contains the common elements between sets A and B. To find A ∩ B, we need to find the elements that appear in both sets:
A = {1, 3, 5, 7}
B = {4, 5, 6}
A ∩ B = {5}
Statement 3: A ⊆ B
The symbol ⊆ represents the subset relationship between sets. A is a subset of B if every element of A is also an element of B. To check whether A is a subset of B, we compare the elements of A with B. If all elements of A are found in B, then A ⊆ B is true. Otherwise, it is false.
A = {1, 3, 5, 7}
B = {4, 5, 6}
Since 1, 3, and 7 are not elements of B, A is not a subset of B. Therefore, A ⊆ B is false.
Summary:
Statement 1: A ∪ B = {1, 3, 4, 5, 6, 7} (True)
Statement 2: A ∩ B = {5} (True)
Statement 3: A ⊆ B = False