Question

A gold vault has 3 locks with a key for each lock. Key A is owned by the

manager whilst Key B and C are in the custody of the senior bank teller

and the trainee bank teller respectively. In order to open the vault door at

least two people must insert their keys into the assigned locks at the same

time. The trainee bank teller can only open the vault when the bank

manager is present in the opening.

i) Determine the truth table for such a digital locking system (4 marks)

ii) Derive and minimize the SOP expression for the digital locking system
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Answer 1

i) Truth Table:

A | B | C | O

0 | 0 | 0 | 0

0 | 0 | 1 | 0

0 | 1 | 0 | 0

0 | 1 | 1 | 0 (condition 2 not satisfied)

1 | 0 | 0 | 0

1 | 0 | 1 | 1 (both conditions satisfied)

1 | 1 | 0 | 1 (both conditions satisfied)

1 | 1 | 1 | 1 (both conditions satisfied)

ii) The minimized sum of products (SOP) expression is

O = AC + AB

Step-by-step explanation:

We have three inputs A, B and C

Let O is the output.

We are given two conditions to open the vault door:

1. At least two people must insert their keys into the assigned locks at the same time.

2. The trainee bank teller (C) can only open the vault when the bank manager (A) is present in the opening.

i) Construct the Truth Table

A | B | C | O

0 | 0 | 0 | 0

0 | 0 | 1 | 0

0 | 1 | 0 | 0

0 | 1 | 1 | 0 (condition 2 not satisfied)

1 | 0 | 0 | 0

1 | 0 | 1 | 1 (both conditions satisfied)

1 | 1 | 0 | 1 (both conditions satisfied)

1 | 1 | 1 | 1 (both conditions satisfied)

ii) SOP Expression using Karnaugh-Map:

A 3 variable Karnaugh-map is attached.

The minimized sum of products (SOP) expression is

O = AC + AB

The orange pair corresponds to "AC" and the purple pair corresponds to "AB"

Bonus:

The above expression may be realized by using two AND gates and one OR gate.

Please refer to the attached logic circuit diagram.