By Eiichi Bannai

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Surjection (or surjective function): an onto function. c 2000 by CRC Press LLC symmetric: the property of a binary relation R that if aRb then bRa. symmetric difference (of relations): for relations R and S on A, the relation R ⊕ S where a(R ⊕ S)b if and only if exactly one of the following is true: aRb, aSb. symmetric difference (of sets): for sets A and B, the set A ⊕ B containing each object that is an element of A or an element of B, but not an element of both. system of distinct representatives: given sets A1 , A2 , .

3. Universal statements in predicate logic are analogues of conjunctions in propositional logic. If variable x has domain D = {x1 , . . , xn }, then (∀x ∈ D)P (x) is true if and only if P (x1 ) ∧ · · · ∧ P (xn ) is true. 4. Existential statements in predicate logic are analogues of disjunctions in propositional logic. If variable x has domain D = {x1 , . . , xn }, then (∃x ∈ D)P (x) is true if and only if P (x1 ) ∨ · · · ∨ P (xn ) is true. 5. Adjacent universal quantifiers [existential quantifiers] can be transposed without changing the meaning of a logical statement: (∀x)(∀y)P (x, y) ⇔ (∀y)(∀x)P (x, y) (∃x)(∃y)P (x, y) ⇔ (∃y)(∃x)P (x, y).

In this case, P is stronger than Q, and Q is weaker than P . Compound propositions P and Q are logically equivalent, written P ≡ Q, P ⇔ Q, or P iff Q, if they have the same truth values for all possible truth values of their variables. A logical equivalence that is frequently used is sometimes called a logical identity. A collection C of connectives is functionally complete if every compound proposition is equivalent to a compound proposition constructed using only connectives in C. A disjunctive normal expression in the propositions p1 , p2 , .

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