Comprehension
Two binary operations \( \oplus \) and \( \) are defined over the set \( \{ a, e, f, g, h \} \) as per the following tables:
Thus, according to the first table \( f \oplus g = a \), while according to the second table \( g h = f \), and so on. Also, let \( f^2 = f \oplus f \), \( g^3 = g g g \), and so on.
| \( \oplus \) | a | e | f | g | h | a |
|---|---|---|---|---|---|---|
| a | a | a | e | f | g | h |
| e | e | e | f | g | h | a |
| f | f | f | g | h | a | e |
| g | g | g | h | a | e | f |
| h | h | h | a | e | f | g |
| \( \) | a | e | f | g | h |
|---|---|---|---|---|---|
| a | a | a | a | a | a |
| e | a | e | f | g | h |
| f | a | f | h | e | g |
| g | a | g | e | h | f |
| h | a | h | g | f | e |
Question: 1
What is the smallest positive integer $n$ such that $g^n = e$?
What is the smallest positive integer $n$ such that $g^n = e$?
Show Hint
When working with binary operations, compute successive powers by applying the operation repeatedly and checking for when the identity element is reached.
Updated On: Aug 1, 2025
- 4
- 5
- 2
- 3
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The Correct Option is D
Solution and Explanation
From the first table, we can compute the powers of $g$. We know that:
\[
g^1 = g, \quad g^2 = g \circ g = f, \quad g^3 = g \circ g \circ g = e
\]
Thus, the smallest positive integer $n$ such that $g^n = e$ is 3.
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Question: 2
Upon simplification, $f \circ f \circ f \circ (f \circ (f \circ f))$ equals:
Upon simplification, $f \circ f \circ f \circ (f \circ (f \circ f))$ equals:
Show Hint
When simplifying expressions with multiple operations, apply the binary operation step by step and refer to the operation table for accurate results.
Updated On: Aug 1, 2025
- e
- f
- g
- h
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The Correct Option is B
Solution and Explanation
From the table for $\circ$ operation, we can simplify step by step:
\[
f \circ f = g, \quad f \circ (f \circ f) = f \circ g = h
\]
Now compute:
\[
f \circ f \circ f = f \circ h = g
\]
Thus, $f \circ f \circ f \circ (f \circ (f \circ f)) = f$.
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Question: 3
Upon simplification, $a^{10} \circ (f \circ (g \circ g)) \circ e^8$ equals:
Upon simplification, $a^{10} \circ (f \circ (g \circ g)) \circ e^8$ equals:
Show Hint
When simplifying complex expressions, follow the order of operations and use the table to determine intermediate values.
Updated On: Aug 1, 2025
- e
- f
- g
- h
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The Correct Option is A
Solution and Explanation
We simplify the expression step by step:
\[
g \circ g = f \quad \text{(from the first table)}
\]
Now we compute:
\[
f \circ f = g \quad \text{and then} \quad a^{10} \circ g = e
\]
Thus, the entire expression simplifies to $e$.
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