Answer :
Let's analyze each statement one by one.
a. All equivalent sets are equal sets.
To determine if this statement is true or false, we need to understand the definitions:
- Equivalent sets: Two sets are said to be equivalent if they have the same number of elements, also called cardinality.
- Equal sets: Two sets are considered equal if they contain exactly the same elements.
Having the same number of elements does not necessarily mean the sets contain the same elements. Therefore, not all equivalent sets are equal sets.
Thus, the statement is False.
b. A set is a subset of itself.
A set [tex]\(A\)[/tex] is a subset of another set [tex]\(B\)[/tex] if every element in [tex]\(A\)[/tex] is also in [tex]\(B\)[/tex]. Since every element in a set [tex]\(A\)[/tex] is, by definition, also in [tex]\(A\)[/tex] itself, the set [tex]\(A\)[/tex] is always a subset of itself.
Hence, the statement is True.
c. [tex]\(\{m, n, o\} \subseteq \{m, n\}\)[/tex]
For a set [tex]\(\{m, n, o\}\)[/tex] to be a subset of [tex]\(\{m, n\}\)[/tex], every element in the set [tex]\(\{m, n, o\}\)[/tex] must be in the set [tex]\(\{m, n\}\)[/tex]. However, the element [tex]\(o\)[/tex] is not in the set [tex]\(\{m, n\}\)[/tex].
Therefore, the statement is False.
d. [tex]\(2 \in \{0, 4, 12\}\)[/tex]
The symbol [tex]\(\in\)[/tex] means "is an element of." So, we are checking whether [tex]\(2\)[/tex] is an element of the set [tex]\(\{0, 4, 12\}\)[/tex].
Since [tex]\(2\)[/tex] is not among the elements [tex]\(0, 4,\)[/tex] or [tex]\(12\)[/tex],
The statement is False.
e. If [tex]\(A \cup B = \emptyset\)[/tex], then [tex]\(A = B = \emptyset\)[/tex].
The union of two sets is empty, if and only if both sets are themselves empty. This is because if either set had any elements, their union would not be empty.
Thus, the statement is True.
f. If [tex]\(A \cap B = \emptyset\)[/tex], then [tex]\(A = B = \emptyset\)[/tex].
The intersection of two sets is the set of elements they have in common. For the intersection to be empty, it simply means there are no common elements, but it does not imply that either set is necessarily empty.
So, the statement is False.
g. [tex]\(A \cap \emptyset = A\)[/tex], for any set [tex]\(A\)[/tex]
The intersection of any set [tex]\(A\)[/tex] with the empty set [tex]\(\emptyset\)[/tex] is always the empty set, because there are no elements that [tex]\(A\)[/tex] and [tex]\(\emptyset\)[/tex] share.
Thus, the statement is False.
h. [tex]\(A \cup \emptyset = A\)[/tex], for any set [tex]\(A\)[/tex]
The union of any set [tex]\(A\)[/tex] with the empty set [tex]\(\emptyset\)[/tex] is always the set [tex]\(A\)[/tex] itself, because there are no additional elements to add from [tex]\(\emptyset\)[/tex].
Therefore, the statement is True.
Summarizing all the answers:
a. False
b. True
c. False
d. False
e. True
f. False
g. False
h. True
a. All equivalent sets are equal sets.
To determine if this statement is true or false, we need to understand the definitions:
- Equivalent sets: Two sets are said to be equivalent if they have the same number of elements, also called cardinality.
- Equal sets: Two sets are considered equal if they contain exactly the same elements.
Having the same number of elements does not necessarily mean the sets contain the same elements. Therefore, not all equivalent sets are equal sets.
Thus, the statement is False.
b. A set is a subset of itself.
A set [tex]\(A\)[/tex] is a subset of another set [tex]\(B\)[/tex] if every element in [tex]\(A\)[/tex] is also in [tex]\(B\)[/tex]. Since every element in a set [tex]\(A\)[/tex] is, by definition, also in [tex]\(A\)[/tex] itself, the set [tex]\(A\)[/tex] is always a subset of itself.
Hence, the statement is True.
c. [tex]\(\{m, n, o\} \subseteq \{m, n\}\)[/tex]
For a set [tex]\(\{m, n, o\}\)[/tex] to be a subset of [tex]\(\{m, n\}\)[/tex], every element in the set [tex]\(\{m, n, o\}\)[/tex] must be in the set [tex]\(\{m, n\}\)[/tex]. However, the element [tex]\(o\)[/tex] is not in the set [tex]\(\{m, n\}\)[/tex].
Therefore, the statement is False.
d. [tex]\(2 \in \{0, 4, 12\}\)[/tex]
The symbol [tex]\(\in\)[/tex] means "is an element of." So, we are checking whether [tex]\(2\)[/tex] is an element of the set [tex]\(\{0, 4, 12\}\)[/tex].
Since [tex]\(2\)[/tex] is not among the elements [tex]\(0, 4,\)[/tex] or [tex]\(12\)[/tex],
The statement is False.
e. If [tex]\(A \cup B = \emptyset\)[/tex], then [tex]\(A = B = \emptyset\)[/tex].
The union of two sets is empty, if and only if both sets are themselves empty. This is because if either set had any elements, their union would not be empty.
Thus, the statement is True.
f. If [tex]\(A \cap B = \emptyset\)[/tex], then [tex]\(A = B = \emptyset\)[/tex].
The intersection of two sets is the set of elements they have in common. For the intersection to be empty, it simply means there are no common elements, but it does not imply that either set is necessarily empty.
So, the statement is False.
g. [tex]\(A \cap \emptyset = A\)[/tex], for any set [tex]\(A\)[/tex]
The intersection of any set [tex]\(A\)[/tex] with the empty set [tex]\(\emptyset\)[/tex] is always the empty set, because there are no elements that [tex]\(A\)[/tex] and [tex]\(\emptyset\)[/tex] share.
Thus, the statement is False.
h. [tex]\(A \cup \emptyset = A\)[/tex], for any set [tex]\(A\)[/tex]
The union of any set [tex]\(A\)[/tex] with the empty set [tex]\(\emptyset\)[/tex] is always the set [tex]\(A\)[/tex] itself, because there are no additional elements to add from [tex]\(\emptyset\)[/tex].
Therefore, the statement is True.
Summarizing all the answers:
a. False
b. True
c. False
d. False
e. True
f. False
g. False
h. True
