Find the Boolean expression of Logic circuit

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The starting point is to establish truth tables for the 8 logic symbols:

 

These can now be applied to the circuits.

A B C NAND-1 NAND-2 AND-1 AND-2 NOR-1 D
0 0 0 1 1 1 0 0 1
0 0 1 1 1 1 0 0 1
0 1 0 1 1 1 1 0 1
0 1 1 1 1 1 1 0 1
1 0 0 1 1 1 0 0 1
1 0 1 1 0 0 0 1 0
1 1 0 0 1 0 1 0 1
1 1 1 0 0 0 0 1 0

This is the main results table with some extra columns for intermediate results (see below).

The circuit diagram shows 3 NAND-gates, 2 AND-gates and 1 NOR-gate. We need to label the gates.

A and B feed NAND-1; A and C feed NAND-2; D is identified as NAND-3; so NAND-1 output=0 only when A=B=1; otherwise it's 1. NAND-1=A NAND B or NAND-1=¬(A∧B).

Similarly A=C=1 make NAND-2 output=0, and any other values make it 1. NAND-2=A NAND C or NAND-2=¬(A∧C).

AND-1 (upmost) =NAND-1 AND NAND-2=1 when NAND-1=NAND-2=1; otherwise 0. AND-1=NAND-1∧NAND-2.

AND-2=B AND NAND-2=1 when B=NAND-2=1; otherwise 0. AND-2=B∧NAND-2.

NOR-1=AND-1 NOR AND-2=1 when AND-1=AND-2=0, otherwise 0. NOR-1=¬(AND-1∨AND-2).

Finally, D=NOR-1 NAND C=0 when NOR-1=C=1; otherwise 1. D=¬(NOR-1∧C).

D=¬(NOR-1∧C)=¬((¬(AND-1∨AND-2))∧C),

D=¬((¬((NAND-1∧NAND-2)∨(B∧NAND-2)))∧C),

D=¬((¬(((¬(A∧B))∧(¬(A∧C)))∨(B∧(¬(A∧C)))))∧C) is the combined Boolean expression.

(See also your other similar question for Boolean notation.)

by Top Rated User (1.1m points)

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