How Does the Absorption Law Simplify This Boolean Function?

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The discussion centers on applying the absorption law to simplify the Boolean function f(x1, x2, x3, x4) = [(not x1) * x2 * x4] ∨ [x2 * x3 * x4]. The absorption law allows for the reduction of complex expressions by absorbing like terms. The teacher's solution simplifies the function to f(x1, x2, x3, x4) = x2 * x3 * x4, which the student initially found confusing. The student later realizes that the absorption occurs because one term covers multiple instances of 1s in a coverage matrix, leading to a clearer understanding of the simplification process. This illustrates the effectiveness of the absorption law in Boolean algebra.
Lilia
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Homework Statement


i'm viewing an example written in class. it looks like this:

f(x1, x2, x3, x4) = [(not x1) * x2 * x4] ∨ [x2 * x3 * x4]

what should be function after applying absorption law?

Homework Equations


i know how another option called "gluing" works:

[x1 * x2 * x3] ∨ [(not x1) * x2 * x3] = x2 * x3

The Attempt at a Solution


our teacher wrote f(x1, x2, x3, x4) = x2 * x3 * x4 but I'm having a hard time to understand why
 
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Lilia said:

Homework Statement


i'm viewing an example written in class. it looks like this:

f(x1, x2, x3, x4) = [(not x1) * x2 * x4] ∨ [x2 * x3 * x4]

what should be function after applying absorption law?

Homework Equations


i know how another option called "gluing" works:

[x1 * x2 * x3] ∨ [(not x1) * x2 * x3] = x2 * x3

The Attempt at a Solution


our teacher wrote f(x1, x2, x3, x4) = x2 * x3 * x4 but I'm having a hard time to understand why

From: https://www.electronics-tutorials.ws/boolean/bool_6.html
  • Absorptive Law – This law enables a reduction in a complicated expression to a simpler one by absorbing like terms.
    • A + (A.B) = A (OR Absorption Law)
    • A(A + B) = A (AND Absorption Law)
Are you sure you typed the question right? The form of your equation doesn't seem to match the OR form of the Absorptive Law...
 
Last edited:
now i see.

we were writing quine-maccluskey algorithm example, and we applied this law because in coverage matrix (quine matrix) there was no column with single 1.

this is the matrix:
-------------------------------------------------
implicant | 0111 | 1100 | 1110 | 1111 |
-------------------------------------------------
01-1 ...|...1...|...|...|...|
-------------------------------------------------
-111 ...|...1...|...|...|...1...|
-------------------------------------------------
111- ...|...|...|...1...|...1...|
-------------------------------------------------
--00 ...|...|...1...|...|...|
-------------------------------------------------
1--0 ...|...|...1...|...1...|...|
-------------------------------------------------
first and second row is my question. but now i think i get it.

since second row covers two 1s, and the first row - one 1, that's why it's called absorption
in the same manner 5th absorbs the 4th one, and the result is [x1 * (not x4)] (as our teacher wrote)

am i right?
 
Last edited:

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