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.
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c) Implication:We say that “p implies q” and writep→q to designate the statement, which is the implication of q by p. Alternatively, we can also say
(i) “If p, then q.” (ii) “p is sufficient for q.”
(iii) “p is a sufficient condition for q.” (iv) “q is necessary for p.”
(v) “q is a necessary condition for p.” (vi) “p only if q.”
A verbal translation of p→q for our example is “If combinatorics is a required course for sophomores, then Margaret Mitchell wrote Gone with the Wind.” The statement p is called the hypothesis of the implication; q is called the conclusion.
When statements are combined in this manner, there need not be any causal relationship between the statements for the implication to be true.
d) Biconditional: Last, the biconditional of two statementsp, q, is denoted byp↔q, which is read “p if and only if q,” or “p is necessary and sufficient for q.” For our p, q, “Combinatorics is a required course for sophomores if and only if
Margaret Mitchell wrote Gone with the Wind” conveys the meaning of p↔q.
We sometimes abbreviate “p if and only if q” as “p iff q.”
Truth Tables, Tautologies, and Logical Equivalence
Mathematics normally works with a two-valued logic: Every