# functionally complete Propositional logics commonly use a set of logical connectives consisting of 5 elements: negation, conjunction, disjunction, implication and bijection, i.e. $${NOT, AND, OR, IMPLY, IFF}$$. Predicate logics commonly use the same symbols with addition of two quantifiers, $${NOT, AND, OR, IMPLY, IFF, EXISTS, FORALL}$$ Although $$IFF$$ connective can be constructed from $$IMPLY$$ conectives, it is often included in the basic set of logical connectives because it lowers complexity of compound formulae; basically, it can be regarded as a mere shorthand. In fact, many connectives can be descibed in term of others. A **functionally complete** set of logical connectives (Boolean operators) is the one which can be used to produce all other logical connectives. Such a set can express all possible truth tables by combining its elements into a Boolean expression. A well-known functionally complete set is $${AND,NOT}$$, consisting of binary conjunction and negation. Each of the singleton sets { NAND } and { NOR } is functionally complete. IMPLY and AND connectives can be described in terms of OR connective - the minimum set of syymbols from which all other symbols can be constructed is --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/300-logic/terms/functional-completness.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.