1. (25 points) Let X be a finite set of positive integers. We say X is good if and only if for every nonempty set Y contained in X, the sum of the members of Y is not divisible by 10. What is the largest number n, such that there is a good set X with exactly n members? Give an example of a good set of that size. Explain why there is no larger
good set.
2. (25 points) Is the conjunction of the following schemata truth-functionally satisfiable?
Explain your answer.
• (p11 ∨ p12) ∧ (p21 ∨ p22) ∧ (p31 ∨ p32)
• p11 ⊃ ¬(p21 ∨ p31)
• p21 ⊃ ¬(p11 ∨ p31)
• p31 ⊃ ¬(p11 ∨ p21)
• p12 ⊃ ¬(p22 ∨ p32)
• p22 ⊃ ¬(p12 ∨ p32)
• p32 ⊃ ¬(p12 ∨ p22)
3. (25 points) How many truth assignments to the sentence letters p1, . . . , p5, q1, . . . , q5 satisfy the following schema?
(p1 ⊃ q1) ∧ . . . ∧ (p5 ⊃ q5)
4. (25 points) Recall that ⊕ represents exclusive disjunction. How many truth assignments to the sentence letters p1, . . . , p5 satisfy the following schema?
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