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| Conditional and Biconditional Statements |
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| It is already mentioned in earlier classes that compound statements of different propositions can be obtained by conjunction, disjunction and negation of propositions. |
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| We just recall these three basic logical connectivities and their truth table and subsequently we shall learn about conditional and biconditional statements. |
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| If p and q are two propositions, then the compound proposition |
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"p and q" is known as conjunction of the proposition. It is denoted by  |
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| [The conjunction of two propositions p and q is true if both p and q are true and in all other cases it is false] |
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The truth table of p  q is given below |
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The last column gives the truth table value of p q. |
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If p and q are two propositions, the compound proposition "p or q" is called the disjunction of p and q. It is denoted by p  q. |
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[The disjunction p  q of two proposition p and q are false if both p and q are false and in all other cases, it is true] |
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The truth table of p  q is given below |
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The last column gives the truth value of p  q. |
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| Let p be any proposition. The proposition "not p" is called the negation of p. It is denoted by ~p. |
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| [The negation of p is false if p is true and the negation of p is true if p is false] |
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| Following is the truth table for ~p. |
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| If p and q are two propositions, then the compound proposition, "if p then q" is known as conditional statement or implication. |
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It is denoted by p  q. |
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| p is called the antecedent (or hypothesis). |
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| q is called the consequent (or conclusion). |
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Note: p  q can be read as |
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| (i) If p then q |
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| (ii) p is sufficient for q |
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| (iii) q if p |
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| (iv) q is necessary for p |
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| (v) p only if q |
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| The truth value of a conditional statement is given by the following rule. |
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[The conditional statement p q is false whenever p is true and q is false, otherwise p q is true.] |
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The following table shows the truth table of the conditional statement p  q. |
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Truth table for the conditional statement p  q is |
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We write each truth value of p  q as follows. |
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| Let us consider the logical statement "If x is prime then x2 is not prime." |
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| Let p: x is a prime number. |
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| q: x2 is not a prime number. |
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Here if p is true and q is true, then p  q is true. That is if x is a prime number, then x2 is not a prime number, for further reference, we state |
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| C1 |
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If both p and q has truth value T, then p  q has truth value T. |
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(T  T = T) |
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| If p has the truth value T, q has the truth value T. |
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| p: x is a prime number |
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| q: x2 is not a prime number |
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p  q has the truth value T |
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| C2 |
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If p has the truth value T and q has the truth value F, then p  q has the truth value F (T  F = F). |
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| If p has the truth value T, q has the truth value F. |
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| p: x is a prime number |
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| q: x2 is a prime number |
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p  q: If x is a prime number then x2 is not a prime number. This statement is logically false p  q has the truth value F. |
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| C3 |
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If p has the truth value F and q has the truth value T, then p  q has the truth value T (F  T = T). |
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| If p has a truth value F and q has the truth value T, then |
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| p: is not a prime number |
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| q: x2 is not a prime number |
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We know that whether x is prime or not, x2 is not a prime number. This implies the truth value of p  q is true. Therefore the truth value of p  q is T. |
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| C4 |
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If both p and q have the truth value F, p  q has the truth value |
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T(F  F = T). |
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| It should be noted that in the above example, the simple statement p and q depend on each other. In other words, if the truth value of one depends on the truth value of the other and |
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p  q is true, then the conditional statement |
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p  q is called implication. |
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Moreover, even though there is no connection between the simple proposition p and q, the conditional p  q is true. For example, |
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| let |
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| p: e is a vowel |
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| q: 3 + 4 = 7 |
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Here the conditional p  q is true because the consequent (q) is true. |
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It should be clear that the conditional statement, p  q has truth value T, whenever the truth value of p is false. |
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