Given the sets [tex]\( A = \{a, b, c\} \)[/tex] and [tex]\( B = \{c, a\} \)[/tex], we need to determine which of the following statements is true:
1. [tex]\( B \subset A \)[/tex]
2. [tex]\( A \subset B \)[/tex]
### Step-by-Step Solution:
1. Check if [tex]\( B \subset A \)[/tex]:
- A set [tex]\( B \)[/tex] is a subset of another set [tex]\( A \)[/tex] if every element of [tex]\( B \)[/tex] is also an element of [tex]\( A \)[/tex].
- Elements of [tex]\( B \)[/tex] are [tex]\( c \)[/tex] and [tex]\( a \)[/tex].
- Check if these elements are present in [tex]\( A \)[/tex]:
- [tex]\( c \in A \)[/tex] (True, since [tex]\( c \)[/tex] is in [tex]\( A \)[/tex])
- [tex]\( a \in A \)[/tex] (True, since [tex]\( a \)[/tex] is in [tex]\( A \)[/tex])
- Since both elements of [tex]\( B \)[/tex] are present in [tex]\( A \)[/tex], [tex]\( B \subset A \)[/tex] is true.
2. Check if [tex]\( A \subset B \)[/tex]:
- A set [tex]\( A \)[/tex] is a subset of another set [tex]\( B \)[/tex] if every element of [tex]\( A \)[/tex] is also an element of [tex]\( B \)[/tex].
- Elements of [tex]\( A \)[/tex] are [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
- Check if these elements are present in [tex]\( B \)[/tex]:
- [tex]\( a \in B \)[/tex] (True, since [tex]\( a \)[/tex] is in [tex]\( B \)[/tex])
- [tex]\( b \in B \)[/tex] (False, since [tex]\( b \)[/tex] is not in [tex]\( B \)[/tex])
- [tex]\( c \in B \)[/tex] (True, since [tex]\( c \)[/tex] is in [tex]\( B \)[/tex])
- Since not all elements of [tex]\( A \)[/tex] are present in [tex]\( B \)[/tex], [tex]\( A \subset B \)[/tex] is false.
### Conclusion:
- The statement [tex]\( B \subset A \)[/tex] is true.
- The statement [tex]\( A \subset B \)[/tex] is false.
Thus, the correct answers are:
- [tex]\( B \subset A \)[/tex] is true.
- [tex]\( A \subset B \)[/tex] is false.
