To determine which statement is true, we need to understand the symbols ⊂ and ⊆.
⊂ means "subset". If set A is a subset of set B, it means that every element in A is also in B. In other words, all the elements in A are contained within B, but B may also have additional elements.
⊆ means "subset or equal". If set A is a subset or equal to set B, it means that every element in A is also in B, and B may have additional elements. In this case, A and B can be the same set.
Now let's evaluate the given statements:
Statement 1: B ⊂ C and C ⊆ A
This means that every element in B is also in C, and every element in C is also in A. Looking at the sets:
B = {l, a, t, e}
C = {t, a, l, e}
A = {l, a, t, e, r}
This statement is true since B is a subset of C and C is a subset of A.
Statement 2: C ⊆ B and D ⊂ B
This means that every element in C is also in B, and D is a proper subset of B (meaning D is a subset of B but not equal to B). Looking at the sets:
C = {t, a, l, e}
B = {l, a, t, e}
D = {e, a, t}
This statement is not true because D is not a proper subset of B.
Statement 3: D ⊂ A and A ⊂ D
This means that every element in D is also in A, and A is a proper subset of D. Looking at the sets:
D = {e, a, t}
A = {l, a, t, e, r}
This statement is not true because A is not a proper subset of D.
Statement 4: B ⊂ A and C ⊂ D
This means that every element in B is also in A, and every element in C is also in D. Looking at the sets:
B = {l, a, t, e}
A = {l, a, t, e, r}
C = {t, a, l, e}
D = {e, a, t}
This statement is not true because C is not a subset of D.
Statement 5: D ⊆ A and A ⊂ C
This means that every element in D is also in A, and A is a proper subset of C. Looking at the sets:
D = {e, a, t}
A = {l, a, t, e, r}
C = {t, a, l, e}
This statement is not true because A is not a proper subset of C.
Therefore, the true statement is:
B ⊂ C and C ⊆ A