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3 Topics ii Valid & Invalid Arguments.pptx.ppt
- 1. Valid & Invalid Arguments Logical Quantifiers and their negation Priority and Precedence Tautologies Contradictions By Prof. Najam Mphil (Edu) BS Hon M.A Edu M.Ed Special Educagion PGD (CS) CCNA MCSE Microsoft Certified system Engineer Mikrotik Certified System Engineer Email-ASSADCHADHAR@GMAIL.COM #-03127522112 Discrete Structure for Computer Science
- 2. Valid & Invalid Arguments oArgument is a sequence of statements ending in a conclusion. oDetermination of validity of an argument depends only on the form of an argument, not on its content. “If you have a current password, then you can log onto the network.” p=“You have a current password” q=“You can log onto the network.” p → q p ∴ q where ∴ is the symbol that denotes “therefore.”
- 3. Valid & Invalid Arguments oAn argument is a sequence of statements, and an argument form is a sequence of statement forms(have proposition var.). o All statements in an argument and all statement forms in an argument form, except for the final one, are called premises (or assumptions or hypotheses). oThe final statement or statement form is called the
- 4. Valid & Invalid Arguments •oTo say that an argument form is valid means that no matter what particular statements are substituted for the statement variables in its premises, if the resulting premises are all true, then the conclusion is also true. Conclusion q is valid, when (p1 ∧ p2 ∧ · · · ∧ pn) → q is a tautology. •oTo say that an argument is valid means that its form is valid.
- 5. Logical Quantifiers and their negation • There are two ways to quantify a propositional function: universal quantification and existential quantification. They are written in the form of “ xp(x)” and “ xp(x)” respectively. To negate a quantified ∀ ∃ statement, change to , and to , and then negate the statement. ∀ ∃ ∃ ∀ • وجودی اور مقدار عالمگیر :ہیں طریقے دو کے کرنے درست مقدار کی فعل تجویزی بالترتیب وہ مقدار۔ "∀xp(x)" اور "∃xp(x)" مقداری ہیں۔ جاتے لکھے میں شکل کی کریں نفی کی بیان پھر اور ،∀ سے ∃ اور ،∃ کو ∀ ،لیے کے کرنے نفی کی بیان Logical Quantifiers and their negation
- 6. priority and precedence Priority and Precedence
- 7. Tautologies • Firstly, here are some examples of tautologies in mathematics: ( p q ) p is a mathematical ∧ ⇒ statement that will always be true and is, therefore, a tautology. In words, this says that if the truth of p and q together is true, then p is true. • مثالیں کچھ کی ٹیوٹولوجی میں ریاضی یہاں ،پہلے سے سب ہیں: ( p ∧ q ) ⇒ p درست ہمیشہ جو ہے بیان ریاضیاتی ایک کہتا یہ ،میں الفاظ ہے۔ ٹاٹولوجی ایک یہ لیے اس اور گا رہے اگر کہ ہے p اور q تو ،ہے درست ساتھ ایک سچائی کی p ہے۔ درست
- 8. Contradictions • In Mathematics, a contradiction occurs when we get a statement p, such that p is true and its negation ~p is also true. Now, let us understand the concept of contradiction with the help of an example. Consider two statements p and q. Statement p: x = a/b, where a and b are co-prime numbers. • بیان ایک ہمیں جب ہے ہوتا وقت اس تضاد ،میں ریاضی p ،ہے ملتا کہ جیسے p نفی کی اس اور ہے درست ~p ہم اب ہے۔ درست بھی ہیں۔ سمجھتے کو تصور کے تضاد سے مدد کی مثال ایک p اور q پر کریں۔ غور a
- 9. Precedence of Logical Operators • As in arithmetic, an ordering is imposed on the use of logical operators in compound propositions • We will generally use parentheses to specify the order in which logical operators in a compound proposition are to be applied. p q r (p) (q (r)) • To avoid unnecessary parenthesis, the following precedences hold: 1. Negation () 2. Conjunction () 3. Disjunction () 4. Implication () 5. Biconditional ()
- 10. Logical Equivalences: • Definition:Compound propositions that have the same truth values in all possible cases are called logically equivalent. • Propositions p and q are logically equivalent if p q is a tautology. • Informally, p and q are equivalent if whenever p is true, q is true, and vice versa • Notation: p q (p is equivalent to q), p q, and p q • Alert: is not a logical connective
- 11. Logical Equivalences: • Are the propositions (p q) and (p q) logically equivalent? • To find out, we construct the truth tables for each: p q pq p pq 0 0 0 1 1 0 1 1 The two columns in the truth table are identical, thus we conclude that (p q) (p q) Example