The modulus function, denoted as \(f(x) = |x|\), returns the absolute value of \(x\). It’s useful in number theory and analyzing distances.
The GCD of two integers is the largest integer that divides both numbers without leaving a remainder. It’s fundamental in number theory.
The LCM of two integers is the smallest positive integer that is divisible by both numbers. It’s commonly used in problems involving synchronization.
Denoted as \(n!\), the factorial of \(n\) is the product of all positive integers up to \(n\). It’s essential in combinatorics and permutations.
Denoted as \(C(n, k)\) or \(\binom{n}{k}\), it represents the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to order.
Denoted as \(P(n, k)\), it represents the number of ways to arrange \(k\) elements from a set of \(n\) elements in a specific order.
Logarithms, especially base 2 (\(\log_2\)), are used in analyzing algorithms, particularly for their time complexity.
Exponentiation involves raising a number to a power, which is critical in calculating growth rates, especially in recursive algorithms.
The signum function, denoted as \(\text{sgn}(x)\), extracts the sign of a real number. It’s useful in various mathematical proofs.
A function that determines whether a number is prime or composite, which is fundamental in number theory.
A discontinuous function that is zero for negative inputs and one for positive inputs. It’s used in discrete event simulation and signal processing.
Denoted as \(\sum\), it represents the sum of a sequence of numbers, essential in combinatorial analysis and series.
Denoted as \(\prod\), it represents the product of a sequence of numbers, useful in various mathematical and combinatorial contexts.
These functions are integral to discrete mathematics, providing tools to analyze, solve, and understand complex problems. Let me know if you need further details or examples for any of these functions!
| #### {Powers} |
|---|
| #### {Exponentials Functions} |
| \[ e^a \times e^b = e^{a+b}\] |
| \[ (e^a )^b = e^{ab}\] |
| %\[ \lfloorx\rfloor\] |
| : |
| State the range and domain of the following function \[ F(x) = \lfloor x-1 \rfloor \] |
| #### Special Cases |
| Anything to the power of zero is always 1 |
| \[ X^ 0 = 1 \mbox{ for all values of X} \] |
| Mathematical Functions{data-navmenu=“Functions”} ===================================== |
| Column {.tabset} |
% http://www.purplemath.com/modules/logs2.htm
The Relationship: “_b(x) = y” means the same thing as “b y = x”. logarithms are really exponents (powers); they’re just written differently.
(b) = 1, for any base b, because b1 = b.
(1) = 0, for any base b, because b0 = 1.
(a) is undefined if a is negative.
(0) is undefined for any base b.
(bn) = n, for any base b.
\(a^x = y\) \(\log_a(y) = x\)
\(e^x = y\) \(ln(y)=x\)
\(\log_a(x\times y) = \log_a(x) + \log_a(y)\)
\(\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)\)
\(\log_a(\frac{1}{x}) = - \log_a(x)\)
\(\log_a(a) = 1\)
\(\log_a(1) = 0\)
Convert \(6^3 = 216\) to the equivalent logarithmic expression.
To convert, the base (that is, the 6)remains the same, but the 3 and the 216 switch sides. This gives me:
\[\log_6(216) = 3 \]
Convert \(\log_4(1024) = 5\) to the equivalent exponential expression.
To convert, the base (that is, the 4) remains the same, but the 1024 and the 5 switch sides. This gives me:
\[4^5 = 1024\]
The following laws are very useful for working with logarithms.
Multiplication inside the can be turned into addition outside the , and vice versa. \[\mbox{\log}_b(mn) = \mbox{\log}_b(m) + \mbox{\log}_b(n)\]
Division inside the can be turned into subtraction outside the , and vice versa. \[\mbox{\log}_b(m/n) = \mbox{\log}_b(m) – \mbox{\log}_b(n)\]
An exponent on everything inside a can be moved out front as a multiplier, and vice versa. \[mbox{\log}_b(m^n) = n \times \mbox{\log}_b(m)\]
\[ \log_b(x) = \frac{1}{\log_x(b)} \]
\[ \log_b(x) = \frac{\log_a(b)}{\log_a(b)} \]
\[ \log_3(x) + 3 \log_x(3) = 4 \]
\[ \left(\log_3(x)\right)^2 + 3 = 4 \log_3(x) \]
\[ \log_3(x) + 3 \log_x(3) = 4 \]
\[ \left(\log_3(x)\right)^2 - 4 \log_3(x) + 3 = 0 \]
\[ e^a \times e^b = e^{a+b}\]
\[ (e^a )^b = e^{ab}\]
\[ \frac{\mbox{ln}(2) + \mbox{ln}(2^2) + \mbox{ln}(2^3) + \mbox{ln}(2^4) + \mbox{ln}(2^5) } {\mbox{ln}(4)} \] Useful Rule of Logarithms \[ \mbox{ln}(a^b) = b\cdot \mbox{ln}(a) \] \[ \frac{\mbox{ln}(2) + 2 \cdot \mbox{ln}(2) + 3 \cdot\mbox{ln}(2) + 4 \cdot \mbox{ln}(2) + 5 \cdot \mbox{ln}(2) } {\mbox{ln}(4)} \] ——————–
In mathematics and computer science, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted floor(x) or ⌊x⌋.
Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ceil(x) or ⌈x⌉
For example, ⌊2.4⌋ = 2, ⌊−2.4⌋ = −3, ⌈2.4⌉ = 3, and ⌈−2.4⌉ = −2.
The integral part or integer part of x, often denoted \([x]\) is usually defined as the ⌊x⌋ if x is nonnegative, and ⌈x⌉ otherwise
For example, [2.4] = 2 and [−2.4] = −2.
The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part.
Some authors define the integer part as the floor regardless of the sign of x, using a variety of notations for this.
For n an integer, \(⌊n⌋ = ⌈n⌉ = [n] n =\) n.
x Floor ⌊x⌋ Ceiling ⌈x⌉ Fractional part {x} 2 2 2 0 2.4 2 3 0.4 2.9 2 3 0.9 −2.7 −3 −2 0.3 −2 −2 −2 0
http://www.analyzemath.com/college_algebra/problems_5.html Let X = {1, 2, 3, 4} and Y={A,B,C,D}. Determine whether each relation is a function, with \(X -> Y\).
No: Two different ordered pairs (2, C) and (2, A) in f have the same number 2 as their first coordinate
Let X = {1, 2, 3, 4}. Determine whether the following relation on X is a function.
No The element 2 does not appear as the first coordinate in any ordered pair in g.
Complete the following table for the functions
\[ \begin{array}{|c||c|c|c|c|c|c|} \hline x & \phantom{p}1\phantom{p}& & & & 81 & \\ \hline \phantom{p} $g(x)$ \phantom{p}& & \phantom{p}1\phantom{p} & \phantom{p}2\phantom{p} & & & \phantom{p}5\phantom{p} \\ \hline \phantom{p}$h(x)$ \phantom{p}& & & & 3.00 & \phantom{p}\phantom{p}\phantom{p} & \\ \hline \end{array} \]
Express your answers to 2 decimal places only.
\[ \begin{array}{|c||c|c|c|c|c|c|} \hline x & 1 & 3 & 9 & 27 & 81 & 243 \\ \hline \phantom{p} g(x) \phantom{p}&0 & 1 & 2 & 3 & 4 & 5 \\ \hline \phantom{p} h(x) \phantom{p}& 1.00 & 1.44 & 2.08 & 3.00 & 4.33 & 6.24 \\ \hline \end{array} \]
The function \(f : \mathbb{Z} \rightarrow \mathbb{Z}\) is defined by the rule: \[f(x) = |x + 1| \] What is the range of this function?
Is this function ``onto” ?
Is this function ``one-to-one” ?
Is this function invertible ?
Compute the Logarithm of the following
Evaluate the following expression. \[ \mbox{Log}_4 64 + \mbox{Log}_5 625 + \mbox{Log}_9 3 \]
State the range and domain of the following function \[ F(x) = \lfloor x-1 \rfloor \]
Let \(S\) be the set of all 4 bit binary strings.
The function \(f : S \rightarrow \mathbb{Z}\) is defined by the rule: \[f(x) = \mbox{the number of zeros in x}\] for each binary string \(x \in S\). Find:
Evaluate \(f(6)\)
\[ f(x) = \lfloor \frac{x+1}{2} \rfloor \]
\[ f(6) = \lfloor \frac{\boldsymbol{6}+1}{2}\rfloor = \lfloor \frac{7}{2}\rfloor\]
\[ \lfloor 3.5 \rfloor = 3 \]
Evaluate \(f(-6)\)
\[ f(x) = \lfloor \frac{x+1}{2} \rfloor \]
\[ f(6) = \lfloor \frac{\boldsymbol{-6}+1}{2}\rfloor = \lfloor \frac{-5}{2}\rfloor\]
\[ \lfloor -2.5 \rfloor = -3 \]
| ### Combined Questions |
| % Section 4 Functions % http://doc.gold.ac.uk/~maa01km/solutions/tut4sol.pdf |
| [(b)] Let \(S\) be the set of all 4 bit binary strings. |
| The function \(f : S \rightarrow \mathbb{Z}\) is defined by the rule: \[f(x) = \mbox{the number of zeros in x}\] for each binary string \(x \in S\).\ Find: |
| a. the number of elements in the domain b. \(f(1000)\) c. the set of pre-images of 1 d. the range of \(f\). |
[(c)]