14 CHAPTER 1. INTRODUCTION obvious from the previous theorem. A Definition is a statement that is true by interpreting one of its terms in such a way as to make the statement true An Ariom or Assumption is a statement that is taken to be true without proof. A Tautology is a statement which is true without assumptions(for np ) A Contradiction is a statement that cannot be true(for example, A is true and A is false) There are other important logical connectives for statements be and“兮”:“∧” means“and”;“V" means“or';and“” means meaning of these connectives is given by a truth table, where"T" stands for a true statement and"F" stands for a false statement. One can consider the truth table as an axiom AB~ AAABJAVBJA→BA F he truth table a is true and i is fals Then n a is false since a is true.aab is false since b is. Vb is true since at least one statement(A)is true, A=B is false since A can't imply B when A is true and B isn't. Notice that if A is false, then a= B is always true since B can be anythin Manipulating these connectives, we can prove some useful tautologies The first set of tautologies are the commutative, associative, and distributive laws. To prove these tautologies, one can simply generate the appropriate truth table. For example, the truth table to prove (AV(BAC)+((avb)a (AVC) is ABCIBAC| AV(BACAVBJAVCI(A∨BA(A以 TITIT TFFFT TTTTTTFF TTTTTFTF TTTTTFFF FFIF F14 CHAPTER 1. INTRODUCTION obvious from the previous theorem. A Definition is a statement that is true by interpreting one of its terms in such a way as to make the statement true. An Axiom or Assumption is a statement that is taken to be true without proof. A Tautology is a statement which is true without assumptions (for example, x = x). A Contradiction is a statement that cannot be true (for example, A is true and A is false). There are other important logical connectives for statements besides ì⇒î and ì⇔î: ì∧î means ìandî; ì∨î means ìorî; and ì∼î means ìnotî. The meaning of these connectives is given by a truth table, where ìTî stands for a true statement and ìFî stands for a false statement. One can consider the truth table as an Axiom. Table 1 A B ∼ A A ∧ B A ∨ B A ⇒ B A ⇔ B T T F T T T T T F F F T F F F T T F T T F F F T F F T T To read the truth table, consider row two where A is true and B is false. Then ∼ A is false since A is true, A ∧ B is false since B is, A ∨ B is true since at least one statement (A) is true, A ⇒ B is false since A canít imply B when A is true and B isnít. Notice that if A is false, then A ⇒ B is always true since B can be anything. Manipulating these connectives, we can prove some useful tautologies. The first set of tautologies are the commutative, associative, and distributive laws. To prove these tautologies, one can simply generate the appropriate truth table. For example, the truth table to prove (A∨(B∧C) ⇔ ((A∨B)∧ (A ∨ C)) is: A B C B ∧ C A ∨ (B ∧ C) A ∨ B A ∨ C (A ∨ B) ∧ (A ∨ C) T T T T T T T T T T F F T T T T T F T F T T T T T F F F T T T T F T T T T T T T F T F F F T F F F F T F F F T F F F F F F F F F