# Logic study problems

### Problem 1

Consider the following claim:

Claim: For every dinosaur D, if D eats meat, then D has sharp teeth and D is not huge.

State the negation and the contrapositive of the claim. Your answer should be in words, with all negations (e.g. "not") on individual predicates.

Write out your solution, actually write it out on paper, before consulting the annotated solution

### Problem 2

(a) For which values of p, q, and r is the following logical expression true? $$(\neg p \vee q) \wedge (q \rightarrow r) \wedge (\neg r \vee p)$$

Give a succinct description of which combinations of input values work, rather than the whole truth table.

(b) Show that the following two expressions aren't logically equivalent: $$(p \rightarrow q) \wedge r$$ $$p \rightarrow (q \wedge r)$$

Write out your solution, actually write it out on paper, before consulting the annotated solution

### Problem 3

Recall that "$$\exists ! x \in A,\ P(x)$$" means that there is exactly one value x in the set A that makes P(x) true.

(a) Express "$$\exists ! x \in A,\ P(x)$$" using the other standard logical operations. You can use any combination of shorthand symbols (e.g. $$\wedge$$) and words (e.g. "and"). Concentrate on capturing the meaning correctly.

(b) State the negation of your answer to (a), moving all instances of "not" onto individual predicates.