A.2 The Logic of Predicates and Quantifiers 595In Exercises 21 - 28 state the negation of the given symbolic statement, and
use the principles of this section to "simplify" where appropriate.- P v '"" Q 25. P ( Q R)
22.PA(QVR) 26 .(PVQ)VR - (PV Q) R 27. (QA P) (RV S)
- P v (Q R) 28. (P Q) * R
In Exercises 29 - 40 translate the given sentence into logical symbols, use
the rules of this section to obtain the negation, and translate the result back
into smooth English.- John is innocent and Mary's charge is a lie.
- If it rains tomorrow there will be no picnic.
- I'll order pizza if you come tonight.
- I like you but I don't like the clothes you wear.
- I am going either to the concert or to the football game.
- Either we win soon or I'll quit the team.
- This statement is true only if I can prove it.
- If x =f. 0, then x^2 > 0.
- The number 1 is neither a prime number nor a composite
number. - f is continuous at xo if f is differentiable at xo.
- If you come to my house for dinner, I'll either grill a steak or
make stir fry. - If you look in the right lo cation tonight you will see the comet
if the sky is clear.
A.2 The Logic of Predicates and Quantifiers
Many statements commonly made in mathematics, such as "f'(x) = x^2 -
5 sin 2x" have the form of a proposition, but are not propositions because they
contain variables. Their truth-values cannot be determined as long as the values
of the variables a re unknown. The variables in the above equation are f and x.
We call such statements "propositional functions,'' or "predicates."
Definition A.2.1 A propositional function (or predicate) is a declarative
sentence containing one or more variables, which becomes a proposition when
the variables are replaced by constants.