2.1 Propositions and Logical Operations

A statement or proposition is a declarative sentence that is either true or false, but not both. Statement examples: The earth is round 2 + 3 = 5 Examples that are not statements: Do you speak Spanish? (this is a question) Take two aspirins. (this is a command)

Propositional variables used in logic are p q and r. These variables can be replaced with statements where p: The sun is shining today. q: It is cold. Statements can be combined by logical connectives to form compound statements. and is represented by ∧ or is repersented by ∨ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - not is not a connective. not is represented by

and: both statements must be true for the result to be or: only one statement must be true in order for the result to be true We use truth tables for our statements:

Truth Table for (p ∧ q) ∨ ( p) 2 variables p and q 22 =4 4 possible outcomes Col 1 Col 2 Col 3 Col 4 Col 5 p q p ∧ q ~ P (p ∧ q) ∨ (~p) T F

1. List all possibilities of p and q in column 1 and 2. 2. Fill in column 3 using the and connective. Both p and q must be true in order for the result to be true. 3. Fill in column 4 . Take the opposite of what is in column 1 and place the result in column 4. 4. Fill in column 5 by comparing column 3 and 4 using the or connective. Only one statement must be true in order for the result to be true.

Negation Statement 2 is an even integer and 8 is an odd integer To negate the statement 2 is not an even integer or 8 is not an odd integer

Universal quantification of P(x) Denoted ∀ x P(x) Meaning: For all values of x, P(x) is either true or false Example: P(x): -(-x)= x P(x) can be called a proposition that results in either true or false The opposite of a negative number is a positive number always. ∀ x P(x) is true (the universal quantification of P(x) is true)

Q(x) = x + 1 < 4 ∀ x Q(x) The universal quantification of Q(x) is false because we can replace x with a number that makes this statement false.

Existential quantification of P(x) is denoted ∃ xP(x) There exists a value(s) of x for which P(x) is true. Example: Q(x) : x + 1 ≺ 4 is true because x could be 1 or 2. ∃y where y + 2 = y is false because there is no value that makes this statement true

In programming we have IF statements or IF THEN ELSE statements. IF N ≺ 10 THEN replace N with N + 1 Return N The N ≺ 10 is called the guard The IF statement is a statement that can be either true or false.