Math Problem Statement

Simplify the Boolean algebra expression Y = f(A, B, C, D) = Σ m(1, 3, 7, 8, 10, 12, 13, 15) and draw the logic circuit for the reduced expression.

Solution

The Boolean expression provided is:

Y=f(A,B,C,D)=Σm(1,3,7,8,10,12,13,15)Y = f(A, B, C, D) = \Sigma m(1, 3, 7, 8, 10, 12, 13, 15)

This is a sum of minterms representation, where the given minterms correspond to specific binary combinations of the variables AA, BB, CC, and DD.

Step 1: Generate the Karnaugh Map (K-Map)

We can simplify this Boolean function by placing these minterms into a 4-variable K-map and applying grouping to simplify the expression.

The binary representations of the minterms are:

  • m(1) = 0001
  • m(3) = 0011
  • m(7) = 0111
  • m(8) = 1000
  • m(10) = 1010
  • m(12) = 1100
  • m(13) = 1101
  • m(15) = 1111

We will now populate the K-map.

AB\CD00011110
000110
010011
111110
101010

Step 2: Simplification Using the K-Map

From the K-map, we can group adjacent 1's:

  • Group 1 (4 cells): m(12), m(13), m(15), m(7) → CC
  • Group 2 (2 cells): m(8), m(12) → ABA \overline{B}
  • Group 3 (2 cells): m(3), m(7)

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Math Problem Analysis

Mathematical Concepts

Boolean Algebra
Karnaugh Maps
Logic Circuit Design

Formulas

Sum of Minterms
Karnaugh Map Simplification

Theorems

Simplification of Boolean Functions

Suitable Grade Level

University/College Level