Question

If \(X = \{\,8^n - 7n - 1 \mid n \in \mathbb{N}\,\}\) and \(Y = \{\,49(n-1) \mid n \in \mathbb{N}\,\}\), then

• A. \(X \subset Y\)
• B. \(Y \subset X\)
• C. \(X = Y\)
• D. Information not sufficient

Hint:

To compare two sets:
Compute a few elements of each set
Check subset relations both ways
If neither holds, the sets are not equal

Answer 1

The Correct Option is D

Step 1: Find some elements of set \(X\). For \(n=1\): \[ 8^1 - 7(1) - 1 = 0 \] For \(n=2\): \[ 8^2 - 7(2) - 1 = 64 - 14 - 1 = 49 \] For \(n=3\): \[ 8^3 - 7(3) - 1 = 512 - 21 - 1 = 490 \] Thus, \[ X = \{0,\,49,\,490,\dots\} \]
Step 2: Find some elements of set \(Y\). For \(n=1\): \[ 49(1-1) = 0 \] For \(n=2\): \[ 49(2-1) = 49 \] For \(n=3\): \[ 49(3-1) = 98 \] Thus, \[ Y = \{0,\,49,\,98,\,147,\dots\} \]
Step 3: Compare the sets.
\(490 \in X\) but \(490 \notin Y\) \(\Rightarrow X \nsubseteq Y\)
\(98 \in Y\) but \(98 \notin X\) \(\Rightarrow Y \nsubseteq X\)
Step 4: Since neither set is a subset of the other and they are not equal, the correct conclusion cannot be determined as (A), (B), or (C).