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Our teacher gave us this assignment.

Steve would like to determine the relative salaries of three coworkers using two facts. First, he knows that if Fred is not the highest paid of the three, then Janice is. Second, he knows that if Janice is not the lowest paid, then Maggie is paid the most. Is it possible to determine the relative salaries of Fred, Maggie, and Janice from TCSS312A 2002 Discrete Structures Autumn what Steve knows? If so, who is paid the most and who the least? Explain your reasoning.

I already know the answer that Fred is the highest paid and Janice is the lowest by using this reasoning.

(1) if Fred is not the highest paid of the three, then Janice is

(2) if Janice is not the lowest paid, then Maggie is paid the most

If the first statement is true, the Janice will be the highest paid and that will contradict the second statement that says Maggie is paid the most.

If the first statement is false, that would mean that Fred is the highest paid and because of that the second statement will be false and Janice will be the lowest paid.

My problem is I can't think of a way to prove this by using truth table, direct proof or indirect proof.

I've tried to prove it by truth table and here it is.

Symbols:

p = Fred is the highest paid q = Janice is the highest paid

r = Maggie is the highest paid s = Janice is the lowest paid

Premises:

[1] ~p → q = If Fred is not the highest paid of the three, then Janice is. (GIVEN)

[2] ~s → r = If Janice is not the lowest paid, then Maggie is paid the most. (GIVEN)

[3] p v q v r = Only one of the workers can be the highest paid.

[4] q → ~s = Janice cannot be both the highest and the lowest paid at the same time

My conclusion is p -> s. But the rows 2 and 4 have true premises and wrong conclusions. What is the wrong part. And if you can please give other formal proof.

Please help me. Thanks~

Code (Text):

P q r s ~p ~q ~r ~s ~p → q ~s → r p v q v r q → ~s p->s

T T T T F F F F T T T F T

T T T F F F F T T T T T F

T F T T F T F F T T T T T

T F T F F T F T T T T T F

T T F T F F T F T T T F T

T T F F F F T T T F T T F

T F F T F T T F T T T T T

T F F F F T T T T F T T F

F T T T T F F F T T T F T

F T T F T F F T T T T T T

F F T T T T F F F T T T T

F F T F T T F T F T T T T

F T F T T F T F T T T F T

F T F F T F T T T F T T T

F F F T T T T F F T T T T

F F F F T T T T F F F T T

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# Need formal proof

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