Common Mathematical Functions in Discrete Mathematics: Floor, Ceiling, and Remainder

Explore frequently used mathematical functions (floor, ceiling, remainder) in discrete mathematics and computer science. This guide defines these functions, provides examples, and illustrates their applications in various computational contexts.

Common Mathematical Functions in Discrete Mathematics

Several mathematical functions are frequently used in discrete mathematics and computer science. These functions operate on discrete values (typically integers or whole numbers).

1. Floor Function

The floor function, denoted ⌊x⌋, finds the greatest integer less than or equal to x (rounding down).

Example: Floor Function

  • ⌊3.5⌋ = 3
  • ⌊-2.4⌋ = -3
  • ⌊3.143⌋ = 3

2. Ceiling Function

The ceiling function, denoted ⌈x⌉, finds the smallest integer greater than or equal to x (rounding up).

Example: Ceiling Function

  • ⌈3.5⌉ = 4
  • ⌈-2.4⌉ = -2
  • ⌈3.143⌉ = 4

3. Remainder (Modulo) Function

The remainder function, written a (mod m), finds the remainder when 'a' is divided by 'm'. Formally, a (mod m) = r, where a = mq + r, 'q' is the quotient, and 0 ≤ r < m.

Example: Remainder Function

  • 35 (mod 7) = 0
  • 20 (mod 3) = 2
  • 4 (mod 9) = 4

4. Exponential Functions

The exponential function with base k, denoted kt, is defined as the product of k multiplied by itself t times. k0 = 1, k-m = 1/km (for positive integers m).

For rational exponents a/b, ka/b = b√ka.

Example: Exponential Function

  • 10³ = 1000
  • 51/2 = √5 ≈ 2.236
  • 3-5 = 1/3⁵ = 1/243

5. Logarithmic Functions

The logarithmic function with base n, denoted logn(x), is the inverse of the exponential function. Formally, if k = logn(x), then nk = x.

logn(1) = 0 because n⁰ = 1.

logn(n) = 1 because n¹ = n.

Example: Logarithmic Function

  • log₂(16) = 4 because 2⁴ = 16
  • log₂(100) ≈ 6.64 because 2⁶ = 64 and 2⁷ = 128
  • log₂(0.001) ≈ -9.97 because 2-10 = 1/1024 ≈ 0.00097

Conclusion

These functions are fundamental tools in discrete mathematics and computer science, used for various calculations and modeling.