No of Function possible
Duration: 8 min
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This educational video lecture explains the mathematical derivation for determining the total number of possible Boolean functions given 'n' Boolean variables. The instructor begins by presenting the problem on a slide, stating that with n binary variables, there are $2^n$ possible input combinations. Since a Boolean function can only output two values (0 or 1), the total number of distinct functions is calculated as $2^{(2^n)}$. The lecturer then generalizes this concept using the notation $x^{(y^z)}$, where x represents the nature of the function, y represents the nature of the variable, and z represents the number of variables. He transitions to a whiteboard to illustrate this with a concrete example using two variables, 'a' and 'b', demonstrating how the 4 input combinations lead to 16 possible functions.
Chapters
0:00 – 2:00 00:00-02:00
The video opens with a slide titled 'No of functions possible'. The instructor poses the question: 'With n-Boolean variables how many different Boolean functions are possible?'. He reads the solution provided on the screen, explaining that n binary variables generate $2^n$ combinations. He notes that a Boolean function has only 2 possible values (0 or 1). Consequently, the total number of different functions possible is $2^{(2^n)}$. He introduces a generalized formula $x^{(y^z)}$ to represent this, defining x as the nature of the function, y as the nature of the variable, and z as the number of variables.
2:00 – 5:00 02:00-05:00
The instructor moves to a whiteboard to visualize the concept. He writes two variables, 'a' and 'b', and lists their 4 possible binary combinations: 00, 01, 10, and 11. He draws two ovals labeled 'A' and 'B' to represent the input and output sets. Inside 'A', he writes 'a', 'b', and '2' to indicate 2 variables. Inside 'B', he writes '1' and '2' to indicate 2 possible output values. He explains that for each of the 4 rows in the truth table, the output can be either 0 or 1. He begins drawing a truth table with columns labeled $f_1, f_2, f_3, f_4$ to show different function mappings, eventually concluding that there are 16 possible functions ($2^4$) for 2 variables.
5:00 – 8:02 05:00-08:02
The instructor reinforces the generalization on the whiteboard. He writes 'nature of function -> 2' and 'nature of variable -> 2'. He writes the formula $2^{(2^n)}$ clearly. He explains the components: x (nature of function) is 2, y (nature of variable) is 2, and z (number of variables) is n. He writes examples like $2^{(2^3)}$ to show how the formula applies to 3 variables, solidifying the derivation of the total number of Boolean functions. He emphasizes that the base is the number of output values and the exponent is the number of input combinations.
The lecture provides a clear, step-by-step derivation of the formula for the number of Boolean functions. It starts with the fundamental property of binary variables ($2^n$ combinations) and combines it with the binary nature of the output (2 values) to arrive at the final formula $2^{(2^n)}$. The instructor effectively uses both slide text and whiteboard diagrams to bridge the gap between abstract formulas and concrete examples, specifically demonstrating the 16 functions possible with 2 variables before generalizing to 'n' variables using the notation $x^{(y^z)}$.