By M. Grosser
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The proposition p ∨ ¬p is a tautology (the law of the excluded middle). 2. The proposition p ∧ ¬p is a self-contradiction. 3. A proof that p ↔ q is logically equivalent to (p ∧ q) ∨ (¬p ∧ ¬q) can be carried out using a truth table: p q T T F F T F T F p↔q ¬p ¬q p∧q ¬p ∧ ¬q (p ∧ q) ∨ (¬p ∧ ¬q) T F F T F F T T F T F T T F F F F F F T T F F T Since the third and eighth columns of the truth table are identical, the two statements are equivalent. 4. A proof that p ↔ q is logically equivalent to (p ∧ q) ∨ (¬p ∧ ¬q) can be given by a series of logical equivalences.
Speciﬁcation: in program correctness, a precondition and a postcondition. statement form: a declarative sentence containing some variables and logical symbols which becomes a proposition if concrete values are substituted for all free variables. , written with no punctuation or extra space between them). strongly correct code: code whose execution terminates in a computational state satisfying the postcondition, whenever the precondition holds before execution. subset of a set S: any set T of objects that are also elements of S, written T ⊆ S.
Conjunctive normal form: for a proposition in the variables p1 , p2 , . . , pn , an equivalent proposition that is the conjunction of disjunctions, with each disjunction of the form xk1 ∨ xk2 ∨ · · · ∨ xkm , where xkj is either pkj or ¬pkj . consequent: in a conditional proposition p → q (“if p then q”) the proposition q (“then-clause”) that follows the arrow. consistent: property of a set of axioms that no contradiction can be deduced from the axioms. construct (or program construct): the general form of a programming instruction such as an assignment, a conditional, or a while-loop.