(a) It is raining; It is snowing (b) 4 + 7 = 11; 2 × 4 = 7 8. Let p be “He is tall” and q be “He is handsome”
Write each of the following statements in symbolic form using p and q (a) He is tall and handsome
(b) He is tall but not handsome (c) He is neither tall nor handsome Answers:
(c)
p q p ∧ q (p ∧ q) ∨ (p ∧ q)
T T T T
T F F F
F T F F
F F F F
(d)
p q p ∨ q ~p (p ∨ q) ∨ ~p
T T T F T
T F T F T
F T T T T
F F F T T
(e)
p q ~p ~q ~p ∨ ~q ~(~p ∨ ~q)
T T F F F T
T F F T T F
F T T F T F
F F T T T F
(f)
p q q ∧ p p ∧ (p ∧ q)
T T T T
T F F F
F T F F
F F F F
(g)
p q p ∧ q ~(p ∧ q) p ∧ ~(p ∧ q)
T T T F T
T F F T T
F T F T F
F F F T T
3. (a) Violets are not blue (b) Delhi is not in America
(or It is not the case that Delhi is in America)
(c) 3 + 3 = 7 7. (a) It is raining and it is snowing
(b) 4 + 7 = 11 and 2 × 4 = 7
The symbol → is used to denote connective “If ... then”
The conditional p → q can also be read:
(a) p only if q (b) p implies q (c) p is sufficient for q (d ) q if p
The conditional p → q has two simple statements p and q connected by “if ... then”
The statement p is called the antecedent and the statement q is called the consequent (or conclusion).
If p is true and q is false, then conditional p → q is false. In other cases p → q is true.
The truth values of p → q are given in Table 1.10.
Table 1.10 Truth table for p → q Example 2: Let p: He is a graduate
q: He is a lawyer then,
p → q: If he is a graduate, then he is a lawyer.
1.7.2 Biconditional
A statement of the form “p if and only if q” is called a Biconditional statement. It is denoted by p q (or by p ↔ q).
A Biconditional statement contains the connective “if and only if ” and has two conditions. If p and q have the same truth value, then p ↔ q is true. The truth values p ↔ q are given in Table 1.11.
Table 1.11 Truth table for p ↔ q
p q p ↔ q
T T T
T F F
F T F
F F T
Example 1: Bangalore is in India, if and only if 4 + 4 = 8.
Example 2: 3 + 3 = 6 if and only if 4 + 3 =7.
1.7.3 Converse, Inverse and Contrapositive Propositions
If p → q, is a conditional statement, then (a) q → p is called its converse (b) ~p → ~q is called its inverse (c) ~q → ~p is called its contrapositive.
The truth values of these propositions are given in Tables 1.12, 1.13 and 1.14, respectively.
Table 1.12 Truth table for the converse of p → q
p q p → q q→ p
T T T T
T F F T
F T T F
F F T T
Table 1.13 Truth table for the inverse of p → q
p q ~p ~q ~p→ ~q
T T F F T
T F F T T
F T T F F
F F T T T
Table 1.14 Truth table for contraposition
Example: Write the contrapositive of the implication
“if it is raining, then I get wet”
Solution: let p: It is raining q: I get wet
then the contrapositive is
~q → ~p: If I do not get wet, then it is not raining.
1.8 WELL FORMED FORMULAS
Statement formulas contain one or more simple statements and some connectives. If p and q are any two statements, then
p ∨ q, (p ∧ q) ∨ (~p), (~p) ∧ q
are some statement formulas derived from the statement variables p and q where p and q called the components of the statement formulas. A statement formula has no truth value. It is only when the statement variables in a statement formula are replaced by definite statements that we get a statement, which has a truth value that depends upon the truth values of the statements used in replacing the variables. A statement formula is a string consisting of variables, parentheses and connective symbols.
A statement formula is called a well formed (w f f ) if it can be generated by the following rules:
1. A statement variable p standing alone is a well formed formula.
2. If p is a wellformed formula, then ~p is a well formed formula.
3. If p and q are wellformed formulas, then (p ∧ q), (p ∨ q), (p → q) and (p ↔ q) are well formed formulas.
4. A string of symbols is a well formed formula if and only if it is obtained by finitely many applications of the rules 1, 2 and 3.
According to the above recursive definition of a well formed formula ~(p ∨ q), (~p ∧ q), (p → (p
∨ q)) are well formed formulas.
A statement formula is not a statement and has no truth values. But if we substitute definite statements in place of variables in given formula we get a statement. The truth value of this resulting statement depends upon the truth values of the statements substituted for the variables, which appears as one of the entries in the final column of the truth table constructed. Therefore the truth table of a well formed formula is the summary of truth values of the resulting statements for all possible assignments of values to the variables appearing in the formula. The final column entries of the truth table of a well formed formula gives the truth values of the formula.
1.9 TAUTOLOGY
A statement formula that is true for all possible values of its propositional variables is called a Tautology.
Example 1: (p ∨ q) ↔ (q ∨ p) is a tautology.
Example 2: p ∨ ~p is a tautology.
1.10 CONTRADICTION
A statement formula that is always false is called a contradiction (or absurdity).
Example: p ∧ ~p is an absurdity.
1.11 CONTINGENCY
A statement formula that can either be true or false depending upon the truth values of its propositional variables is called a contingency.
Example: (p → q) ∧ (p ∧ q) is a contingency.
1.12 LOGICAL EQUIVALENCE
Two propositions P and Q are said to be logically equivalent or simply equivalent if P → Q is a tautology.
Example: ~(p ∧ q) and ~p ∨ ~q are logically equivalent.
Two formulas may be equivalent, even if they do not contain the same variables. Two statement formulas P and Q are equivalent if P Q is a tautology and conversely, if P Q is a tautology then P and Q are equivalent. If “P is equivalent Q” then we can represent the equivalence by writing “P ⇔ Q” which can also be written as P ⇔ Q. The symbol “⇔ is not a connective. We usually drop the” quotation marks.
1.13 SOLVED EXAMPLES
Example 1: The converse of a statement is given. Write the inverse and the contrapositive statements
“if I come early, then I can get the car”.
Solution: Inverse: “If I cannot get the car, then I shall not come early”
Contrapositive: If I do not come early, then I cannot get the car.
Example 2: The inverse of a statement is given. Write the converse and contrapositive of the statement.
“If a man is not a fisherman, then he is not a swimmer”.
Solution: Converse: “If he is a swimmer, then the man is a fisherman”.
Contrapositive: “If he is not a swimmer, then the man is not a fisherman”.
Example 3: Determine a truth table of ~p → (q → p)
Solution:
Table 1.15
p q ~p q → p ~p → (q → p)
T T F T T
T F F T T
F T T F F
F F T T T
Example 4: Show that p ∧ ~p is a contradiction.
Solution: The truth table for p ∧ ~p is given below:
Table 1.16
p ~p p ∧ ~p
T F F
T F F
p ∧ ~p is always false, hence p ∧ ~p is a contradiction.
Example 5: Show that p ∨ ~p is a tautology.
Solution: We construct the truth table for (p ∨ ~p) Table 1.17
p ~p (p ∨ ~p)
T F T
T T T
p ∨ ~p is always true.
Hence p ∨ ~p is a tautology.
Example 6: Show that (p ∧ q) → p is tautology.
Solution: Let us construct the truth table for the statement (p ∧ q) → p Table 1.18
p q p ∧ q (p ∧ q) → p
T T T T
T F F T
F T F T
F F F T
In Table 1.18, we notice that the column (4) has all its entries as T. Hence (p ∧ q) → p is a tautology.
Example 7: Show that ~(p → q) ≡ ( p ∧ ~q)
Solution: Let us construct the truth table for the given propositions:
Table 1.19
p q p→ q ~(p → q) ~q p ∧ ~q
T T T F F F
T F F T T T
F T T F F F
F F T F T F
From the truth table it is clear that the truth values of ~(p → q) and p ∧ ~q are identical.
Hence ~(p → q) ≡ p ∧ ~q.