Answer :
To solve the problem, we need to determine whether the set [tex]\(\{b, c\}\)[/tex] is a subset of the set [tex]\(A=\{a, b, c\}\)[/tex], and if it is, in what manner. Here is a detailed step-by-step explanation:
1. Definition of Subset:
A set [tex]\(B\)[/tex] is a subset of set [tex]\(A\)[/tex] if every element of [tex]\(B\)[/tex] is also an element of [tex]\(A\)[/tex]. Mathematically, [tex]\(B \subseteq A\)[/tex] means [tex]\( \forall x (x \in B \implies x \in A)\)[/tex].
2. Check the Elements:
- The elements of the set [tex]\(\{b, c\}\)[/tex] are [tex]\(b\)[/tex] and [tex]\(c\)[/tex].
- We need to verify if both [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are elements of the set [tex]\(A=\{a, b, c\}\)[/tex].
- Clearly, both [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are in [tex]\(A\)[/tex].
3. Proper Subset:
- A proper subset is a subset that is not equal to the parent set. Mathematically, [tex]\(B\)[/tex] is a proper subset of [tex]\(A\)[/tex] if [tex]\(B \subseteq A\)[/tex] and [tex]\(B \neq A\)[/tex].
- In this case, [tex]\(\{b, c\} \subseteq \{a, b, c\}\)[/tex] and [tex]\(\{b, c\} \neq \{a, b, c\}\)[/tex], so [tex]\(\{b, c\}\)[/tex] is a proper subset of [tex]\(A\)[/tex].
4. Evaluate Given Statements:
- First Statement: "True because [tex]\(\{b, c\}\)[/tex] is a subset of [tex]\(A\)[/tex]."
This statement is accurate as [tex]\(\{b, c\}\)[/tex] contains elements that are all in [tex]\(A\)[/tex].
- Second Statement: "True because [tex]\(\{b, c\}\)[/tex] is an element of [tex]\(A\)[/tex]."
This statement is false. [tex]\(\{b, c\}\)[/tex] is not an element of [tex]\(A\)[/tex]; it is a subset.
- Third Statement: "False because [tex]\(A\)[/tex] is not a subset of [tex]\(\{b, c\}\)[/tex]."
This is irrelevant to the question.
- Fourth Statement: "True because [tex]\(\{b, c\}\)[/tex] is a proper subset of [tex]\(A\)[/tex]."
This is true, as shown above.
- Fifth Statement: "True because [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are elements of [tex]\(A\)[/tex]."
This statement is accurate. It confirms the subset relationship.
- Sixth Statement: "False because [tex]\(\{b, c\}\)[/tex] is not equal to [tex]\(A\)[/tex]."
This statement does not address the subset issue directly but emphasizes they are not equal.
- Seventh Statement: "False because [tex]\(\{b, c\}\)[/tex] is not equivalent to [tex]\(A\)[/tex]."
Similar to the previous point, it emphasizes equality, not subset.
- Eighth Statement: "False because [tex]\(A\)[/tex] is not a proper subset of [tex]\(\{b, c\}\)[/tex]."
This is irrelevant to the question.
5. Conclusion:
We conclude that the correct understanding here is:
- [tex]\(\{b, c\}\)[/tex] is indeed a subset of [tex]\(A\)[/tex].
- Furthermore, [tex]\(\{b, c\}\)[/tex] is a proper subset of [tex]\(A\)[/tex].
Hence, the answer to the question "True or False? [tex]\(\{b, c\} \subset A\)[/tex]" is True if we consider the statement: "True because [tex]\(\{b, c\}\)[/tex] is a proper subset of [tex]\(A\)[/tex]".
1. Definition of Subset:
A set [tex]\(B\)[/tex] is a subset of set [tex]\(A\)[/tex] if every element of [tex]\(B\)[/tex] is also an element of [tex]\(A\)[/tex]. Mathematically, [tex]\(B \subseteq A\)[/tex] means [tex]\( \forall x (x \in B \implies x \in A)\)[/tex].
2. Check the Elements:
- The elements of the set [tex]\(\{b, c\}\)[/tex] are [tex]\(b\)[/tex] and [tex]\(c\)[/tex].
- We need to verify if both [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are elements of the set [tex]\(A=\{a, b, c\}\)[/tex].
- Clearly, both [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are in [tex]\(A\)[/tex].
3. Proper Subset:
- A proper subset is a subset that is not equal to the parent set. Mathematically, [tex]\(B\)[/tex] is a proper subset of [tex]\(A\)[/tex] if [tex]\(B \subseteq A\)[/tex] and [tex]\(B \neq A\)[/tex].
- In this case, [tex]\(\{b, c\} \subseteq \{a, b, c\}\)[/tex] and [tex]\(\{b, c\} \neq \{a, b, c\}\)[/tex], so [tex]\(\{b, c\}\)[/tex] is a proper subset of [tex]\(A\)[/tex].
4. Evaluate Given Statements:
- First Statement: "True because [tex]\(\{b, c\}\)[/tex] is a subset of [tex]\(A\)[/tex]."
This statement is accurate as [tex]\(\{b, c\}\)[/tex] contains elements that are all in [tex]\(A\)[/tex].
- Second Statement: "True because [tex]\(\{b, c\}\)[/tex] is an element of [tex]\(A\)[/tex]."
This statement is false. [tex]\(\{b, c\}\)[/tex] is not an element of [tex]\(A\)[/tex]; it is a subset.
- Third Statement: "False because [tex]\(A\)[/tex] is not a subset of [tex]\(\{b, c\}\)[/tex]."
This is irrelevant to the question.
- Fourth Statement: "True because [tex]\(\{b, c\}\)[/tex] is a proper subset of [tex]\(A\)[/tex]."
This is true, as shown above.
- Fifth Statement: "True because [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are elements of [tex]\(A\)[/tex]."
This statement is accurate. It confirms the subset relationship.
- Sixth Statement: "False because [tex]\(\{b, c\}\)[/tex] is not equal to [tex]\(A\)[/tex]."
This statement does not address the subset issue directly but emphasizes they are not equal.
- Seventh Statement: "False because [tex]\(\{b, c\}\)[/tex] is not equivalent to [tex]\(A\)[/tex]."
Similar to the previous point, it emphasizes equality, not subset.
- Eighth Statement: "False because [tex]\(A\)[/tex] is not a proper subset of [tex]\(\{b, c\}\)[/tex]."
This is irrelevant to the question.
5. Conclusion:
We conclude that the correct understanding here is:
- [tex]\(\{b, c\}\)[/tex] is indeed a subset of [tex]\(A\)[/tex].
- Furthermore, [tex]\(\{b, c\}\)[/tex] is a proper subset of [tex]\(A\)[/tex].
Hence, the answer to the question "True or False? [tex]\(\{b, c\} \subset A\)[/tex]" is True if we consider the statement: "True because [tex]\(\{b, c\}\)[/tex] is a proper subset of [tex]\(A\)[/tex]".
