Special chapters
A proposition (or statement or assertion) is a linguistic formation that has the property of being either true or false (principle of the excluded middle). “True” and “false” are called the truth values of the proposition and are denoted by T and F, respe
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Special chapters
4.1
Sets, relations, functions
4.1.1
Basic concepts of mathematical logic
4.1.1.1
Expressions of propositional logic
A proposition (or statement or assertion) is a linguistic formation that has the property of being either true or false (principle of the excluded middle). "True" and" false" are called the truth values of the proposition and are denoted by T and F, respectively. Examples: "5 is a prime number", "3 is a divisor of 7", "3 + 5 = 9". The first of these propositions has the truth value T, the other two F. The equation" 2 + x = 4" is not a proposition, because it has a truth value only after the value of x has been specified (see 2.4.1.2 and 2.4.1.3). Once that is done, it becomes a proposition. Such an expression is called a propositional form or predicate (see 4.1.1.3). If A, and A2 are propositions, then they can be grammatically combined to form new propositions: "not A,", "A, and A 2", "A, or A 2" "if A" then A 2", "A, if and only if A 2", whose truth values depend only on those of the partial propositions occuring in them (principle of extensionality of propositional logic). The truth value of the compound proposition is determined by the classical truth functions non(Atl: "not A,", et(A" A 2): "A, and A 2", vel(A" A 2): "A, or A," in the nonexclusive sense, that is "A, or A2 or both", seq(A, , A 2): "if A, then A,", aeq(A" A2 1: "A, if and only if A2'"
These functions are defined by the following truth tables:
A,
non
A,
A2
e/
vel
seq
aeq
T F
F T
T T F F
T F T F
T F F F
T T T F
T F T T
T F F T
Example: If A, has the truth value F and A2 has the truth value T, then the compound propositions (in the given order) have the values T, F, T, T, F. Further propositions can be formed by compound propositions (for example, et(non(vel(A" A 2 ), seq(A" A 2))). To represent and investigate such propositions one introduces propositional expressions, which can be defined similarly to arithmetical expressions (see 2.4.1.1). Fundamental symbols are constants: T, F; propositional variables: p, , P2, P3, ... ; unary operations: - (sometimes denoted by --,); binary operations: /\, v, -+, +-+; technical symbols (,). 1. Every sequence of symbols consisting of only a propositional variable or a constant is an expression. 2. If H, and H2 are expressions then so are - H" (H, /\ H 2), (H, v H 2), (H, -+ H 2), and (H, +-+ H 2 ).
3. A sequence of symbols is an expression only if it is so by reason of I. or 2. Examples: H, = «p, -+ P2) /\ - P3), H2 = «p, /\ - P3) /\ -(PI /\ P2))' An expression that begins with the symbol - is called a negation; if H == (H, 0 H 2 ), where H, and H2 are expressions and 0 is one of the operations /\, v, -+ or +-+, then H is called a conjunction, disjunction, implication, or equivalence, respectively. As with arithmetic expressions, one introduces a hierachy of separation in which any operation in the sequence +-+,-+,V,I\, """"
I. N. Bronshtein et al., Handbook of Mathematics © Springer Fachmedien Wiesbaden 1979
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4.1.1.1
4.1 Sets, relations, functi
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