Solution to Problem 2'2

1
Solution to Problem 2.2

• ECS 101

2
Problem 2.2

• An island has two clans Truth-tellers (always
  told the truth), and Liars (always lied).
• Silas lived on the island for six years, but
  during all that time he never learned how to
  visually tell the difference between members of
  the two clans. One day Silas met two trolls.
• The first claimed that they belonged to different
  clans,
• Whereas the second proclaimed that the first was
  a liar.
• To which clan did each of the two belong?

3
Identify the Statements

• A statement should have a truth value.
• You should be able to determine if it is true or
  false.
• We find two statements from this problem
• q The first claimed that they belonged to
  different clans.
• p The second proclaimed that the first was a
  liar

4
Finding Relationships in our Problem

• Given
• q The first claimed that they belonged to
  different clans.
• p The second proclaimed that the first was a
  liar.
• Either
• (q and p) or (q and p) is true.
• (q and p) says q is truth and p is lie.
• (q and p) says q is lie and p is truth.
• and
• q is true.
• q says that q is a lie.
• The symbolic expression is (p q) (p q) q

5
Truth Alternatives

• Either
• q is true, p is true, and both trolls are
  Truthtellers.
• q is true, p is false, and the 1st troll is a
  Truthteller and the 2nd is a Liar.
• q is false, p is true, and the 2nd troll is a
  Truthteller and the 1st is a Liar.
• q is false, p is false, and the both trolls are
  Liars

6
Test Logic with Truth Alternative

• q is true, p is true, and both trolls are
  Truthtellers.
• Then
• (q and p) or (q and p) should be true.
• And
• q should be true.
• Clearly neither of these are true.

7
Test Logic with Truth Alternative

• q is true, p is false, and the 1st troll is a
  Truthteller and the 2nd is a Liar.
• Then
• (q and p) or (q and p) should be true.
• And
• q should be false.
• Clearly the first is true, and the second is
  true.
• A solution. Should I stop?

8
Test Logic with Truth Alternative

• q is false, p is true, and the 2nd troll is a
  Truthteller and the 1st is a Liar.
• Then
• (q and p) or (q and p) should be false.
• And
• q should be true.
• Clearly the first of these is false and the
  second of these is true.

9
Test Logic with Truth Alternative

• q is false, p is false, and both trolls are
  Liars.
• Then
• (q and p) or (q and p) should be false.
• And
• q should be false.
• Clearly neither of these are true.

10
Our Solution

• For
• q The first claimed that they belonged to
  different clans.
• p The second proclaimed that the first was a
  liar
• The first is true and the second is false, thus
• First Troll is a Truthteller, and
• Second troll is a Liar.

11
Evaluating Symbolic Expression
So we could have skipped all the alternatives
just by solving the expression.