The Mathematics of Tenure, Car Lights, and Logic Gates

In a state office, an employee maybe tenured if (1) he has passed the Career Service Examination, or (2) he has worked as a probationary employee for more than 10 years.

Notice that the employee maybe tenured if either one of the conditions above is satisfied. He may also be tenured if both conditions are satisfied. If we create a table and rephrase the statements above to question form, we can see that the only reason for an employee not to be tenured is if the answer to both questions is NO.

tenure-table

The first table is also the same as the table shown below.  In the second table,  the situation describes the relationship between a car’s ceiling light and its two doors. Unless the ceiling light is turned on or off manually, it will always be ON if both doors or one of the two doors is open. The only time that the light will be OFF is if the two doors are shut.

car-light-table

The Mathematics 

The operations on the tables above are equivalent to the operations done in  logic gates.  A logic gate performs a logical operation on one or more logic inputs and produces a single output. In the first table, our inputs A and B  are the answer to the two questions, while the output Q is the state of tenure. In the second table, the inputs are the state of the doors (whether they are OPEN or CLOSED), and the output is the state of the light (whether it is ON or OFF).

or logic gate

The logic gate shown in the third figure is an example or OR gate, the gate that operates the two tables. If we replace YES/OPEN/ON with 1 and NO/CLOSED/OFF with 0, then we have a truth table for OR gates.

In set theory, the connective OR is a logical disjunction that results to true whenever one of the operands is true. It is represented by a Venn diagram such as the one shown below. If the circle on the left is A and the circle on the right is B, then A or B means A, B, or both.

Logic gates are now primarily used in electronics particularly in computers. A modern microprocessor, for example, contains millions of logic gates. Aside from OR gate, there are also other gates such as AND gate,  NOT gate, and more.

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