2nd Stage, Lecture 1: Foundation and Introduction to Geometry

Foundation and Introduction to Geometry

1.1. What is a statement (or a proposition)?
A statement is a meaningful sentence that is either true or false (but not both). For example, the number 7 is an odd integer and the number 10 is an even integer.

1.2. The Logic Laws
1. Identity: A thing is itself.
2. Excluded Middle: A statement is either true or false.
3. Non contradiction: No statement is both true and false.

1.4. Simple and Compound Statements: A statement may be simple or compound. A compound statement (opposite to simple statement) is formed when two or more simple statements are connected in one sentence.

There are four basic connectives:
1. Conjunction: And ?
2. Disjunction: Or ?
3. Conditional: If-then ?
4. Equivalence: If and only if ?

Necessary and Sufficient Conditions:
? The conditional statement: If p, then q is often stated in the form q is a necessary condition for p. Or p is a sufficient condition for q ? The equivalence statement: p if and only if q is often stated in the form p is both necessary and sufficient condition for q or q if and only if p is often stated in the form q is both necessary and sufficient condition for p.