Introduction to Logic Concepts

Introduction to Logic

• A proposition is a closed sentence that is true or false but not both.
• Open sentences are sentences whose truth cannot be determined due to variable elements.
• Basic logical connectives include negation (~), conjunction (^), disjunction (v), and conditional (->).

Properties of Conjunctions/Disjunctions

• Associative Property: (p ^ q) ^ r = p ^ (q ^ r) for conjunction, and similar for disjunction.
• Commutative Property: p ^ q = q ^ p for conjunction, and similar for disjunction.
• Distributive Property: p ^ (q v r) = (p ^ q) v (p ^ r) and p v (q ^ r) = (p v q) ^ (p v r).
• De Morgan's Laws: ~(p ^ q) = ~p v ~q and ~(p v q) = ~p ^ ~q.

Conditional Statements

• p (hypothesis/antecedent) and q (conclusion/consequent) are parts of conditional statements.
• The inverse: ~p -> ~q
• The converse: q -> p
• The contrapositive: ~q -> ~p

Biconditionals

• A biconditional is a conjunction of a condition and its converse (p -> q) ^ (q -> p) or p ↔ q.
• Every definition is a biconditional.
• A tautology is always true.
• A contradiction is always false.

Laws of Logic

• Law of the Excluded Middle: p or not p
• Law of Contradiction: p and not p cannot both be true
• Law of Double Negation: ~ (~p) = p
• Law of Simplification: if p and q, then p
• Law of Disjunctive Addition: If p, then p or q
• Law of Disjunctive Elimination: if p or q and not q is true, then p must be true.
• Law of Transposition: if p then q is the same as, if not q then not p.
• Law of Modus Ponens (detachment): If p then q and p is true, then q is true
• Law of Modus Tollens: If p then q and not q are true, then not p must be true
• Law of Syllogism: if p then q and if q then r, then if p then r

Writing Logic Proofs

• Two-column proofs are used in logic proofs: statements and reasons.
• Information such as givens and what needs proving should be given
• Cite previous statement used to support new statement in parentheses.
• Write a conclusion after the final step
• Every statement in a logic proof should be true.

Introduction to Geometry

• A point is a position in space, represented by a dot.
• Points are denoted with capital letters.
• A line has no width and can be extended infinitely in either direction.
• A plane is a flat surface that extends infinitely in two dimensions.
• Lines, points, and segments that are on the same plane are coplanar.

First Postulates

• Basic geometric axioms (postulates) are foundational statements of geometry.
• Points determine lines; lines contain points.
• Distance is a unique positive number between points.
• Segment equality can be asserted.
• Angles are measurable, using degrees.
• Rays and angles can be constructed.

Properties of Equality and Congruence

• Reflexive Property: Every value is equal to itself; Likewise, every segment/angle is congruent to itself .
• Transitive Property: If a = b and b = c, then a = c; Likewise, if segment a is congruent to segment b and segment b is congruent to segment c, then segment a is congruent to segment c.
• Symmetric Property: If a = b, then b = a; Likewise, if segment a is congruent to segment b, then segment b is congruent to segment a.