Question

determine whether \\(subseteq\\), \\(subset\\), both, or neither can be placed in the blank to form a true statement.\\(a = \\{x | x \in \mathbf{n}\text{ and } 5 < x < 10\\}\\)\\(b = \\{x | x \in \mathbf{n}\text{ and } 2 \leq x \leq 9\\}\\)\\(a\underline{quad}b\\)\\(\\)\\(\\)choose the correct answer below.\\(\bigcirc\\) only \\(subset\\)\\(\bigcirc\\) only \\(subseteq\\)\\(\bigcirc\\) both \\(subseteq\\) & \\(subset\\)\\(\bigcirc\\) none of the above

Explanation:

Step1: List elements of set A

$A = \{6, 7, 8, 9\}$ (since $\mathbb{N}$ is natural numbers, so integers >5 and <10)

Step2: List elements of set B

$B = \{2, 3, 4, 5, 6, 7, 8, 9\}$ (natural numbers ≥2 and ≤9)

Step3: Check subset ($\subseteq$)

All elements of A are in B, so $A \subseteq B$ is true.

Step4: Check proper subset ($\subset$)

B has elements not in A (2,3,4,5), so $A \subset B$ is true.

Answer:

both $\subseteq$ & $\subset$