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. Determine the truth table for such a digital locking system

Derive and minimize the canonical SOP expression for the digital locking system

Draw the logic circuit for the minimized system (ii) above electronic

Explanation:

a) The truth table has 3 inputs: A, B, C, considered to be True when the respective key owner has their key inserted into the lock. The 1 output, "opens" is considered to be True when the conditions are met for the vault to be opened.

The truth table is shown in the first attachment.

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b) The SOP expression for "opens" can be derived by the use of a Karnaugh map representation of the truth table. That is shown in the second attachment. The terms of the table can be covered by the minimal sum ...

  Opens = AB + AC

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c) The logic circuit required to directly implement the sum of products (SOP) expression consists of two And gates (one for each product) and one Or gate (to do the sum). The circuit is shown in the third attachment.

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Comment on the circuit

Of course, the circuit could consist of a single Or gate together with a single And gate. The Or gate inputs would be B or C, and the And gate inputs would be A and 'Or-gate output'.