DHARM104 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE
The sentences 1, 2, 3, 4 and 5 are statements that have assigned the truth value either
true/false on there context. For example sentence 5 is a statement and has assign value true if
it will rain today and false if there will no rain today. The sentences 6 and 7 are command
sentences so they are not considered as statement as per the definition above.
Since, we admit only two possible truth values for a statement therefore our logic is
sometimes called a dual-value logic.
As we mention above, throughout the text we shall use capital letters A, B, .....X, Y, Z to
represent the statements in symbolic logic.
For example,
Statement (1): Delhi is capital.
Symbolic logic: D
Here the statement definition ‘Delhi is capital’ is represented by the symbol‘D’. Conse-
quently symbol ‘D’ corresponds to the statement ‘Delhi is capital’. That is, the truth value of
the statement (2) is the truth value of the symbol‘D’.
Statement (2): Hockey is our national game.
Symbolic Logic: H
The statement (2) is represented by the symbolic logic ‘H’ that is, the truth value of ‘H’
is the truth value of the statement (2).
Since, compound statements are formed by use of operator’s conjunction, disjunction,
negation and implication. These operators are equivalent to our everyday language connec-
tives such that ‘and’, ‘or’, ‘not’ and ‘if-then’ respectively.
Conjunction (AND/∧∧∧∧∧)
Let A and B are two statements, then conjunction of A and B is denoted as A ∧ B (read as “A
and B”) and the truth value of the statement A ∧ B is true if, truth values of both the state-
ments A & B are true. Otherwise, it is false.
These conditions of the conjunction are specified in the truth table shown in Fig. 5.3.
ABA ∧ ∧ ∧ ∧ ∧ B
FFF
FTF
TFF
TTT
Fig. 5.3 (Truth table for conjunction)
Conjunction may have more than two statements and by definition it returns true only
if all the statements are true. Consider the example,
Statement (1) : Passengers are waiting (symbolic logic) P
Statement (2) : Train comes late. (symbolic logic) T
Using conjunction connective we obtain the compound statement,
‘Passengers are waiting “and” train comes late’
Given statement can be equally written in symbolic logic as, P ∧ T