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| Conditional and Biconditional Statements (Contd...) |
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| Two propositions (simple or compound) are said to be logically equivalent if they have identical truth values. |
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If p is logically equivalent to q, we denote p  q. |
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We have already constructed the truth table of (~p) q in earlier class. Recall the same truth table. |
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Truth Table for (~p) q |
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Note that the last column of above table and last column of earlier table are identical. This implies p  q is logically equivalent to (~p) q. |
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| C5 |
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| We explain C5 with the following example. |
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| Consider the statement "either the monsoon break in June or the rivers are dried" |
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| Let p: The monsoon does not break in june |
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| q: rivers are dried |
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| Clearly 'if p then q' is true if p is true and q is true. That is, the conditional statement of the above statement is "If the monsoon does not break in June then rivers are dried". |
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This statement is equivalent to "The monsoon breaks in the month of June or the rivers are dried" which is symbolically written as (~p) q. |
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| We know from C5 that |
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| Taking negation on both sides, we have |
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| Using Demorgan's law to L.H.S, we have |
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| Thus we define |
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| Negation of Conditional Statement |
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| C6 |
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| Example: |
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| Write the negation of the conditional statement |
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| "If the weather is cold then it will snow" |
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| The negation of the given statement is "The weather is cold and it will not snow" |
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| Contrapositive of a Conditional Statement |
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| We know that |
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| C7 |
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| The conditional statement is equivalent to it contrapositive or |
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| Example: |
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| Consider the conditional statement "If a quadrilateral is a square, then all the sides are equal". |
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| The contrapositive of this statement is "If all the sides of a quadrilateral are not equal, then it is not a square". |
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If two simple statements p and q are connected by the connective 'if and only if', then the resulting compound statement is called the biconditional statement. Symbolically it is represented by p  q. |
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| Example: An integer is even if and only if it is divisible by 2. It is a biconditional having the truth value T. |
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The biconditional statement p  q is true when either both p and q are true or both p and q are false. |
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| The truth table for biconditional is given below |
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| For further reference, we write the truth values of biconditional statements as follows. |
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| B1: |
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p q has the truth value T if both p and q have the same truth value. |
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| That is |
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p is true and q is true  p  q is true. |
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p is false and q is false  p  q is true. |
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| B2 |
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p q has truth value F if p and q has opposite truth values. |
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| That is |
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If p has truth value T and q has the truth value F, p q has truth value F. |
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| Also, |
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if p has the truth value F and q has the truth value T, p  q has truth value F. |
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| Another way of defining biconditional statement is given below. |
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| B3 |
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If p and q are propositions, then the conjunction of conditionals p  q and q  p is called a biconditional proposition. |
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Let us construct the truth table of (p  q)  (q  p) |
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| The last column of above table is identical to the last column of earlier table. |
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