# logical equivalence examples without truth tables

Solution 2. To illustrate the logical form of arguments, we use letters of the alphabet (such as p, q, and r) to represent the component sentences of an argument. Prove the second of De Morgan's laws and the two distributive laws … Use the truth tables method to determine whether the formula ’: p^:q!p^q is a logical consequence of the formula : :p. Solution. Exercise 2.7. Exercise 1: Use truth tables to show that ~ ~p ” p (the double negation law) is valid. - Use the truth tables method to determine whether p! We can use these equivalences to finally do mathematical proofs. • truth table method and • by the logical proof method (using the tables of logical equivalences.) demonstrated logical equivalence. Statements and Truth Tables. Truth Tables, Tautologies, and Logical Equivalence Mathematics normally works with a two-valued logic : Every statement is either True or False . See tables 7 and 8 in the text (page 25) for some equivalences with conditionals and biconditionals. That is, we can show that equivalences are correct, without drawing a truth table. Also, if you feel you need more practice with truth tables, prove these laws using truth tables. Propositional Logic, Truth Tables, and Predicate Logic (Rosen, Sections 1.1, 1.2, 1.3) TOPICS • Propositional Logic • Logical Operations Proving Equivalences. Note: Any equivalence termed a “law” will be proven by truth … For example: Two plus two equals ve. Do the second of the distributive laws similarly. Pick a couple of those and prove them with a truth table. The following tables summarize those rules. EXERCISES 3-1. equivalence. 1.2. p q :q p!q :(p!q) p^:q T T F T F F T F T F T T F T F T F F F F T T F F Since the truth values for :(p!q) and p^:qare exactly the same for all possible combinations of truth values of pand q, the two propositions are equivalent. Logical Rules of Reasoning: At the foundation of formal reasoning and proving lie basic rules of logical equivalence and logical implications. For example: Two plus two equals four. Solution 1. Examples Find the truth tables for the following statement forms: 1 p_˘q 2 p _(q ^r) 3 (p _q)^(p _r) ... 2.1 Logical Equivalence and Truth Tables 4 / 9. Examples Examples (de Morgan’s Laws) 1 We have seen that ˘(p ^q) and ˘p_˘q are logically equivalent. Build a truth table containing each of the statements. You can use truth tables to determine the truth or falsity of a complicated statement based on the truth or falsity of its simple components. Note that all of those rules can be proved using truth tables. p q :p p^:q p^q p^:q!p^q T T F F T T T F F T F F F T T F F T F F T F F T j= ’since each interpretation satisfying psisatisﬁes also ’.] Exercise 2: Use truth tables to show that pÙ T ” p (an identity law) is valid. 2 Show that ˘(p _q) ˘p^˘q. A Statement (or Proposition) is a sentence that is true or false but not both.