By Smith D., Eggen M., Andre R.

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Because the form of this statement is ∼ (P ∧ Q), where P is “x is even” and Q is “x is prime,” we may deduce that “x is not even or x is not prime,” which has form ∼P ∨ ∼Q. We have applied the replacement rule, using one of De Morgan’s Laws. 2 is essential. The replacement rule allows you to use definitions in two ways. First, if you are told or have shown that x is odd, then you can correctly state that for some natural number k, x = 2k + 1. You now have an equation to use. Second, if you need to prove that x is odd, then the definition gives you something equivalent to work toward: It suffices to show that x can be expressed as x = 2k + 1, for some natural number k.

Let A(x) be an open sentence with variable x. 2 (a). 2 (a) is false. 2 (b). x) A (x) is equivalent to (Ex)[A(x) ∧ (∀y)(A(y) ⇒ x = y)]. x) A (x). ૺ 12. ” (b) Write the symbolic form of the statement of the Mean Value Theorem. ” x→a (d) Write a useful denial of each sentence in parts (a), (b), and (c). 13. x) P (x)? (a) (∀x)P(x) ∨ (∀x) ∼ P(x). (b) (∀x) ∼P(x) ∨ (Ey)(Ez)(y = z ∧ P( y) ∧ P(z)). (c) (∀x)[P(x) ⇒ ( Ey)(P(y) ∧ x = y)]. ૺ (d) ∼ ( ∀x)(∀y)[(P(x) ∧ P( y)) ⇒ x = y]. 4 14. Riddle: What is the English translation of the symbolic statement ∀E E ∀?

3. 4. Determine precisely the hypotheses (if any) and the antecedent and consequent. Replace (if necessary) the antecedent with a more usable equivalent. Replace (if necessary) the consequent by something equivalent and more readily shown. Beginning with the assumption of the antecedent, develop a chain of statements that leads to the consequent. Each statement in the chain must be deducible from its predecessors or other known results. As you write a proof, be sure it is not just a string of symbols.