The Proof Page

by D. A. Kouba

Section 1.1- Propositions and Connectives; Truth Tables

In the study of mathematics, one often must begin with definitions and assumptions (rules of the game), which are assumed to be true. The process of deductive reasoning can then lead to logical conclusions (lemmas, theorems, and corollaries), which need to be proven true. In general, unique or uniform mathematical language and symbols are used to simplify this process. The Proof Page will assist those in the transition from ``how to" mathematics to ``why" mathematics.

: A is a sentence that is either true (T) or false (F).

: Determine which of the following sentences are propositions. Find solutions HERE .

- 1.) The numbers 3 and 11 are prime.

- 2.) The sun rises in the West and sets in the East.

- 3.)

- 4.)

- 5.) If , then

- 6.) If , then

- 7.) Barry Bonds will hit at least 60 homeruns this baseball season.

- 8.) Buy a carton of milk on the way home today.

- 9.) You will buy a carton of milk on the way home today.

- 10.) Bill Clinton was a popular president.

- 11.) Where are the children ?

- 12.) Beethoven was born on a Tuesday.

: Let and be propositions.

1.) The of and is written (read `` and "). The conjunction is true exactly when both and are true.

2.) The of and is written (read `` or "). The disjunction is true exactly when at least one of and is true.

3.) The of is written (read ``not "). The negation is true exactly when is false.

NOTE : The preceding three definitions of logical connectives and their values of true (T) and false (F) are designed to be consistent with standard spoken and written usage of ``and", ``or", and ``not" in the English language.

: A is an expression involving finitely many logical symbols (connectives) and letters (propositions).

: The expression is a propositional form.

The following truth tables illustrate all possible truth values for the propositional forms , , and .

On The Proof Page, in particular, and in advanced mathematics, in general, we are often concerned with the equivalence of mathematical statements, that is, the equivalence of propositional forms. Equivalent forms give us the opportunity to choose, from among all the equivalent forms, a form which is simplest to use or easiest to understand. Propositional forms are

: Determine if the following propositional forms are equivalent.

- 1.)
and
.............Find solution HERE .

- 2.)
and
.............Find solution HERE .

- 3.) and .............Find solution HERE .

NOTE : (ORDER OF OPERATIONS) If parantheses are not used to clearly indicate the use of the connectives, then invoke the connectives in the following order- , , . For example, is equivalent to .

: Provide parantheses to clarify the following ambiguous expressions. Find solutions HERE .

- 1.)

- 2.)

- 3.)

: A is a propositional form for which all of its values are true (T). A is a propositional form for which all of its values are false.

: Truth tables can be used to easily verify the following two statements. The form is a tautology. The form is a contradiction.

: A of a proposition is any proposition which is equivalent to .

: Let proposition be ``Integer is even." Each of the following statements is a denial of , that is, each statemnt is equivalent to .

: Write two denials for each proposition. Find solutions HERE .

- 1.) Integer is odd.

- 2.) You are beautiful and you drive a fast car.

- 3.) He is tired or hungry.

- 4.) Some dogs chase cats.

- 5.) All men love to watch football.

- 6.) If you are a woman, then you are good at calculus.

- 7.) If you are a college student, then you eat pizza and play video games.

RETURN to The Proof Page .

Please e-mail your comments, questions, or suggestions to D. A. Kouba at
kouba@math.ucdavis.edu .