By Knut Mørken
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2) is always true except in the case when all the variables are F. This means that we can in fact remove the brackets and simply write p ∧ q ∧ r, p ∨ q ∨ r ∨ s, without any risk of misunderstanding since it does not matter in which order we evaluate the expressions. Many other mathematical operations, like for example addition and multiplication of numbers, also have this property, and it therefore has its own name; we say that the operators ∧ and ∨ are associative. The associativity also holds when we have longer expressions: If the operators are either all ∧ or all ∨, the result is independent of the order in which we apply the operators.
We then clearly have p(2) = F and p(4) = T. All the usual relational operators like <, >, ≤ and ≥ can be used in this way. The function p(a) := (a = 2) has the value T if a is 2 and the value F otherwise. Without the special notation for assignment this would become p(a) = (a = b) which certainly looks rather confusing. 2. In the context of logic, the values true and false are denoted T and F, and assignment is denoted by the operator :=. A logical statement is an expression that is either T or F and a logical function p(a) is a function that is either T or F, depending on the value of a.
In normal speech we also routinely link such logical statements together with words like ’and’ and ’or’, and we negate a statement with ’not’. Mathematics is built by strict logical statements that are either true or false. Certain statements which are called axioms, are just taken for granted and form the foundation of mathematics (something cannot be created from nothing). Mathematical proofs use logical operations like ’and’, ’or’, and ’not’ to combine existing statements and obtain new ones that are again either true or false.