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Determine whether the premises “If I do not leave my home early or get stuck in a traffic jam, I will be late to my class and get scolded by my teacher”, “If I am late to my class, I will miss the attendance for the day”, and “I have gotten my attendance today” lead to the conclusion “Therefore, I have left my home early today”. Explain which rules of inference are used for each step.
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Let,

H = I leave my home early.

T = I get stuck in traffic.

C = I will be late to my class.

P = I will get scolded by my teacher.

A = I have gotten my attendance for the day.

Based on the given statements we can construct the Premises and Conclusion as follows:

P1: (~H or T) → (C and P).

P2: C → ~A

P3: A

C1: H

Now Premises are always true so:

  1. P3: A is true implies A = true,
  2. Using this in P2 we get C → ~(true) => C → false, Now in order for this Premise to be true C has to false or otherwise if C is true => True → false = false. Therefore so far we got A = true and C = false.
  3. Using the above info in P1 we get (~H or T) → ( false and P). This has to evaluate to True, Now RHS of implication will be false which leads to (~H or T) → false. So (~H or T) cannot be true as the Premise would compute to false, so (~H or T) = false. This is possible only when both ~H and T are false, so we get T = false and ~H = false => H = true which is the conclusion C1.

Therefore we can say that the conclusion is a valid one based on the given premises.

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