Consider the following logic diagram of a combinational circuit where A, B, and

Question

Consider the following logic diagram of a combinational circuit where A, B, and C are inputs and Q is the output. Three 2-input AND gates and two 2-input OR gates are used in the circuit. It is possible to reduce some of the logic gates without changing the functionality of the circuit. Such component reduction results in higher operating speed (less delay time from input signal transition to output signal transition), less power consumption, less cost, and greater reliability. Construct a logic diagram of a circuit which does have the same function output with only two logic gates (instead of five). Please show the steps. [8 marks]

b) Using basic Boolean algebra identities for Boolean variables A, B and C, prove that ABC+ ABC' + AB'C + A'BC = AB + AC + BC. Please show all steps and mention the identities used.

Explanation / Answer

a)image not loaded

b)proof

Introduce ABC+ABC into the LHS(ie;ABC+ABC'+AB'C+A'BC) since we know that ABC+ABC=ABC by Idempotent Law (A+A=A)

we get ,

= ABC+ABC+ABC+ABC'+AB'C+A'BC (IDEMPOTENT LAW)

= ABC+ABC'+ABC+AB'C+ABC+A'BC (REARRANGING)

= AB(C+C')+AC(B+B')+BC(A+A') (taking common variables from each two terms)

= AB+AC+BC (INVERSE LAW , A+A'=1)

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