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 (1 point) Responses true true false
False.
To determine whether each statement is true or false, we need to compare the sets A and B.
Statement 1: A ⊆ B (A is a subset of B)
To check if A is a subset of B, we need to verify if every element in A is also in B. Since 1, 3, and 7 in A are not present in B, the statement is false.
Statement 2: B ⊆ A (B is a subset of A)
To check if B is a subset of A, we need to verify if every element in B is also in A. Since all elements in B (4, 5, 6) are present in A, the statement is true.
Statement 3: A ∩ B = ∅ (the intersection of A and B is the empty set)
The intersection of A and B is the set of elements that are common to both A and B. In this case, the only common element is 5. Since the intersection is not empty, the statement is false.
Therefore, the correct responses to each statement are:
Statement 1: False
Statement 2: True
Statement 3: False
To determine whether each statement is true or false, we need to compare the elements in set A and set B with the given values.
Statement 1: A ∩ B ≠ ∅ (i.e., A intersected with B is not an empty set)
To find the intersection of two sets, A and B, we need to identify the elements that are common to both sets.
A = {1, 3, 5, 7}
B = {4, 5, 6}
A ∩ B = {5} (since 5 is the only element common to both sets).
Since A ∩ B is not an empty set, the statement A ∩ B ≠ ∅ is true.
Statement 2: |A ∩ B| = 1 (i.e., the cardinality of A intersected with B is equal to 1)
The cardinality of a set refers to the number of elements in the set. To determine the cardinality of A ∩ B, we count the number of elements in their intersection.
A ∩ B = {5} (as determined earlier)
|A ∩ B| = 1 (since there is only one element in the intersection).
Since the cardinality of A ∩ B is 1, the statement |A ∩ B| = 1 is true.
Statement 3: A ∩ B ⊆ A (i.e., A intersected with B is a subset of A)
For this statement, we need to check if all the elements in A ∩ B are also present in A.
A = {1, 3, 5, 7}
A ∩ B = {5} (as determined earlier)
All the elements in A ∩ B ({5}) are also present in A.
Therefore, A ∩ B is a subset of A, and the statement A ∩ B ⊆ A is true.
So, the answers to the statements are as follows:
Statement 1: True
Statement 2: True
Statement 3: True