Deductive reasoning - from statements accepted as true, logically draw conclusions. If the theory is not compatible with reality, then the assumptions must indeed be untrue and the theory does not apply.

Propositions and Connectives

Definition - a statement that describes a mathematical object or idea in terms of previously defined items or axioms. These are "creations" of the one who makes the definition and are not subject to proof.

For each of the following, write out the definition:
• Rational Number - page 1:
• Inequalities - page 2:
• Absolute Value - page 6:
• Ordered Pair - page 10:
• Circle - page 13:
• Solution point - page 17:
• Slope - page 25:
• Perpendicular:
• Function - page 34:
• Even function - page 41:
• Odd function - page 41:
• Radian measure for an angle - page 46:
Proposition - a sentence that is either true or false
For each of the following, determine whether the proposition is true or false:
• 2+3 = 6
• My name is John
• What is the reason you took this class? (Not a proposition.)
• x + 3 = 6 (This is a conditional proposition.)
• This sentence is false. (This is an example of a paradox.)

For each of the following, indicate the appropriate definition which needs to be applied. Where appropriate, utilize these (true) propositions to solve some problems from the book.

• The square root of two is not rational.
• One may find the solution set for an inequality by finding where it is an equality. (page 5)
• |x|² = x²
• |x| = sqrt(x²)
• |x| < k if and only if -k < x < k
• Distance formula - page 11
• Standard equation of a circle - page 13
• Test for symmetry - page 20
• Point-slope equation for a line - page 26
• Slope-intercept equation for a line - page 28
• Two lines are parallel if and only if their slopes are equals
• Two lines are perpendicular if and only if their slopes are negative reciprocals of each other.
• If a graph intersects a vertical line at more that one place, the graph is not of a function
• sin²(t) + cos²(t) = 1
• sin(t) is an odd function and cos(t) is an even function
Symbolic logic - representing statements with variables

Connectives:

• conjunction
• "and"
• P ^ Q is true exactly when both P and Q are true
• disjunction
• "or"
• P v Q is true exactly when at least one of P or Q is true
• negation
• "not"
• ~P is true exactly when P is false
Compound proposition - a proposition consisting of simple propositions and connectives

Two propositions are equivalent if and only if they have the same truth value

A Tautology is a compound proposition which is always true.

• P v ~P
• (~P ^ Q) v (P ^ Q)
A Contradiction is a compound proposition which is the negation of a tautology.

Truth Tables - listing all the possible input values and finding the corresponding truth value of the compound proposition

• ~P
• P v Q
• P ^ Q
HOMEWORK: Write the propositions listed earlier in this document in symbolic form. Notice how many of them utilize similar symbolic forms. Make up some compound statements from real life. Write them using symbolic logic. Determine the truth table for each.

Determine truth tables for the following:

1. P ^ ~P
2. P ^ ~Q
3. P ^ P
4. ~P ^ ~Q
5. P ^ (Q v R)
6. (P ^ Q) v (P ^ R)

IF/THEN and IF AND ONLY IF

IF/THEN - Conditional: P => Q means "If P, then Q"

• P => Q is true unless P is true and Q is false
• Equivalent to ~P V Q
• If you make an A on all exams, your final grade will be an A. When is this statement false?
• A falsity (a lie) can be expected to lead to any conclusion.
• Show the following is a tautology:   [(P =>Q) => P] => P
Converse of P =>Q is the statement Q => P

Contrapositive of P => Q is the statement ~Q => ~P

IF AND ONLY IF - Biconditional: P <=> Q mean "P if and only if Q"

• True exactly when P and Q have the same truth values.
• P <=> Q is equivalenet to (P => Q) ^ (Q => P)
HOMEWORK: Verify the following statements by using truth tables.
1. Theorem: P => Q is equivalent to its contrapositive
2. Theorem: P => Q is not equivalent to its converse