The number of binary operations on a set with 5 elements is greater than 1018 . (A) true (B) false

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Define Binary Operation: A binary operation on a set with 5 elements can be thought of as a function from the Cartesian product of the set with itself (5x5 = 25 possible input pairs) to the set (5 possible outputs for each input pair). To find the total number of binary operations, we need to calculate the number of functions from a set of 25 elements to a set of 5 elements.Calculate Total Functions: Each of the 25 input pairs can be mapped to any of the 5 elements in the set. Since each mapping is independent, the total number of binary operations is 5 raised to the power of 25.Exponential Calculation: Calculate 5^25 using the property of exponents. This is a large number, so we can use logarithms or a calculator to approximate if necessary. However, for the purpose of comparison to 10^18, we can reason about the magnitude of the number.Comparison with 10^18: We know that 5^25 is much larger than 5^18 because raising a number greater than 1 to a higher power will always result in a larger number. Since 5 is greater than 1, 5^25 is greater than 5^18.Evaluate 5^25 vs 10^18: Now, we compare 5^18 to 10^18. Since 5^18 is clearly less than 10^18 (because 5 < 10), we need to determine if the additional power of 7 (from 5^18 to 5^25) is enough to make 5^25 greater than 10^18.Utilize Exponent Properties: To compare 5^25 to 10^18, we can use the fact that 10^18 = (5 * 2)^18 = 5^18 * 2^18. We want to know if multiplying 5^18 by 2^18 makes it smaller or larger than 5^25.Final Comparison: Since 2^18 is a large number, and we are comparing it to 5^7 (the difference between 5^25 and 5^18), we can see that 2^18 is much smaller than 5^7. This is because 2^3 is 8, and 5^3 is 125, and 125 is significantly larger than 8. Therefore, 5^7, which is 5 raised to a power more than twice that of 3, is much larger than 2^18.Conclusion: Therefore, 5^25 is indeed greater than 10^18, and the statement ＂The number of binary operations on a set with 5 elements is greater than 10^18＂ is true.

The number of binary operations on a set with $5$ elements is greater than $10^{18}$ . $\newline$ (A) true $\newline$ (B) false

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The number of binary operations on a set with $5$ elements is greater than $10^{18}$ . $\newline$ (A) true $\newline$ (B) false