I need answers to a,b,c and d. Thanks! Prepositional wffs and truth tables belon

Question

I need answers to a,b,c and d. Thanks!

Prepositional wffs and truth tables belong to a system of two-valued logic because everything has one of two values, false or true. Three-valued logic allows a third value of null or "unknown" (Section 4.3 discusses the implications of three-valued logic on databases). The truth tables for this three-valued system follow. Viewing N as "unknown " explain why it is reasonable to define T N = N, F N = N. and N' = N. Suppose that the statement "Flight 237 is on time" is true, the statement "Runway conditions are icy" is false, and the truth value of the statement "Flight 51 is on lime" is unknown. Find the truth values of the following statements: Runway conditions are not icy and flight 51 is on time. Flight 51 is on time and flight 237 is not Right 51 is not on time or runway conditions are not icy.

Explanation / Answer

From the truth table we can infer that the operator ^ does a logical AND operation and 'v' does logical OR.

a) when a statement's value is unknown, we don't know whether it's true or false.

As one of the statements has unknown value and is ANDed with a true statement, the resulting statement cannot be asserted as either true or false because we don't know what the value of N is. So it only makes sense to take the value of resulting statement as unknown too. So, T^N=N. Note that F^N=F. This is because a false statement logically ANDed with any statement results in a false statement because part of the final statement is false. But when we take T^N, one of the statements is true and the AND value depends on the value of the other statement.In fact, when we AND a true statement with any other statement, the resulting statement will have the value of the other statement.

FvN=N.We are performing logical OR of a false statement and a statement of unknown value. Normally, when we say "either this or that" we are saying the truth if any of this or that are true.In an OR operation if one of the statements is false, then the value of the compound statement depends on the value of the other statement and if the other statement is amiguous, so is the compound statement.

N' = N. This is the complementing operation. When the value of a statement is unknown, it is impossible to assert its complement to be true or false and hence its complement also has unknown value.

b) This statement has two parts.The statement "Runway conditions are not icy" is true since "Runway conditions are icy" is false. The statement "Flight 51 is on time" is amiguous. These two sentences are connected by AND and we have T^N = N. This statement therefore has Unknown value.

c)"Flight 51 is on time" -->N. "Flight 237 is not" --> F. N^F = F. The compound statement is False.

d)"Flight 51 is not on time" ---> N' = N. "Runway conditions are not icy" --->T. N or T = T. Therefore the compound statement is True.

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