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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Q. The number of binary operations on a set with

5

elements is greater than

10^{18}

\newline

(A) true

\newline

(B) false

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 ( $5\times5 = 25$ 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 $10^{18} = (5 \times 2)^{18} = 5^{18} \times 2^{18}$ . 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 $5^7$ $3$ raised to a power more than twice that of $5^7$ $4$ , 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.