Math 221 Notes

Basic Connectives and Truth Tables
In the development of any mathematical theory, assertions are made in the form of sentences.
Such verbal or written assertions, called statements (or propositions), are declarative sentences that are either true or false—but not both. For example, the following are statements, and we use the lowercase letters of the alphabet (such as p, q, and r) to represent these statements. p: Combinatorics is a required course for sophomores. q: Margaret Mitchell wrote Gone with the Wind. r: 2 + 3 _ 5.
The preceding statements represented by the letters p, q, and r are considered to be primitive statements, for there is really no way to break them down into anything simpler.
New statements can be obtained from existing ones in two ways.
1) Transform a given statement p into the statement ￢p, which denotes its negation and is read “Not p.”
For the statement p above, ￢p is the statement “Combinatorics is not a required course for sophomores.” (We do not consider the negation of a primitive statement to be a primitive statement.)
2) Combine two or more statements into a compound statement, using the following logical connectives.
a) Conjunction: The conjunction of the statements p, q is denoted by p ∧ q, which is read “p and q.” In our example the compound statement p ∧ q is read “Combinatorics is a required course for sophomores, and Margaret Mitchell wrote Gone with the Wind.”
b) Disjunction: The expression p ∨ q denotes the disjunction of the statements p, q and is read “p or q.” Hence “Combinatorics is a required course for sophomores, or Margaret Mitchell wrote Gone with the Wind” is the verbal translation for p ∨ q, when p, q are as above.We use the word “or” in the inclusive sense here.
Consequently, p ∨ q is true if one or the other of p, q is true or if both of the statements p, q are true. In