2 ~ Theory of Computer Science
So the sentence 4 is a proposition. For the same reason, the sentences 5 and
6 are propositions. To sentences 7 and 8, we cannot assign truth values as they
are not declarative sentences. The sentence 9 looks like a proposition.
However, if we assign the truth value T to sentence 9, then the sentence asserts
that it is false. If we assign the truth value F to sentence 9, then the sentence
asserts that it is true. Thus the sentence 9 has either both the truth values (or
none of the two truth values), Therefore, the sentence 9 is not a proposition,
We use capital letters to denote propositions,
1.1.1 CONNECTIVES (PROPOSITIONAL CONNECTIVES
OR LOGICAL CONNECTIVES)
Just as we form new sentences from the given sentences using words like
'and', 'but', 'if', we can get new propositions from the given propositions
using 'connectives'. But a new sentence obtained from the given propositions
using connectives will be a proposition only when the new sentence has a truth
value either T or F (but not both). The truth value of the new sentence
depends on the (logical) connectives used and the truth value of the given
propositions.
We now define the following connectives. There are five basic
connectives.
(i) Negation (NOT)
(ii) Conjunction (AND)
(iii) Disjunction (OR)
(iv) Implication (IF
THEN ,:~/
(v) If and Only If.
Negation (NOT)
If P is a proposition then the negation P or NOT P (read as 'not PO) is a
proposition (denoted by -, P) whose truth value is T if P has the truth value
F, and whose truth value is F if P has the truth value T. Usually, the truth
values of a proposition defined using a connective are listed in a table called
the truth table for that connective (Table 1.1),
TABLE 1.1 Truth Table for Negation
p
T
F
,p
F
T
Conjunction (AND)
If P and Q are two propositions, then the conjunction of P and Q (read as 'P
and Q') is a proposition (denoted by P 1\ Q) whose truth values are as given
in Table 1.2.
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So the sentence 4 is a proposition. For the same reason, the sentences 5 and
6 are propositions. To sentences 7 and 8, we cannot assign truth values as they
are not declarative sentences. The sentence 9 looks like a proposition.
However, if we assign the truth value T to sentence 9, then the sentence asserts
that it is false. If we assign the truth value F to sentence 9, then the sentence
asserts that it is true. Thus the sentence 9 has either both the truth values (or
none of the two truth values), Therefore, the sentence 9 is not a proposition,
We use capital letters to denote propositions,
1.1.1 CONNECTIVES (PROPOSITIONAL CONNECTIVES
OR LOGICAL CONNECTIVES)
Just as we form new sentences from the given sentences using words like
'and', 'but', 'if', we can get new propositions from the given propositions
using 'connectives'. But a new sentence obtained from the given propositions
using connectives will be a proposition only when the new sentence has a truth
value either T or F (but not both). The truth value of the new sentence
depends on the (logical) connectives used and the truth value of the given
propositions.
We now define the following connectives. There are five basic
connectives.
(i) Negation (NOT)
(ii) Conjunction (AND)
(iii) Disjunction (OR)
(iv) Implication (IF
THEN ,:~/
(v) If and Only If.
Negation (NOT)
If P is a proposition then the negation P or NOT P (read as 'not PO) is a
proposition (denoted by -, P) whose truth value is T if P has the truth value
F, and whose truth value is F if P has the truth value T. Usually, the truth
values of a proposition defined using a connective are listed in a table called
the truth table for that connective (Table 1.1),
TABLE 1.1 Truth Table for Negation
p
T
F
,p
F
T
Conjunction (AND)
If P and Q are two propositions, then the conjunction of P and Q (read as 'P
and Q') is a proposition (denoted by P 1\ Q) whose truth values are as given
in Table 1.2.
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