1. Logical Operators (Connectives) We will examine the following logical operators: Negation (NOT,  ) Negation (NOT,  ) Conjunction (AND,  ) Conjunction (AND,  ) Disjunction (OR,  ) Disjunction (OR,  ) Exclusive-or (XOR,  ) Exclusive-or (XOR,  ) Implication (if – then,  ) Implication (if – then,  ) Biconditional (if and only if,  ) Biconditional (if and only if,  ) Truth tables can be used to show how these operators can combine propositions to compound propositions.

2. 2 Negation (NOT) Unary Operator, Symbol:  P  P P P P true (T) false (F) true (T)

3. 3 Conjunction (AND) Binary Operator, Symbol:  PQ P QP QP QP Q TTT TFF FTF FFF Note: p ∧ q is T if and only if p is T and q is T.

4. 4 Disjunction (OR) Binary Operator, Symbol:  PQ P  Q TTT TFT FTT FFF Note: p V q is F is and only if q is F and q is F.

5. 5 Exclusive Or (XOR) Binary Operator, Symbol:  PQ PQPQPQPQ TTF TFT FTT FFF P  Q : Today is either Tuesday or it is Thursday. : Pat is either male or female only one of P or Q must be true

6. Implication (continued) –Equivalent forms: If P, then QIf P, then Q P implies QP implies Q If P, QIf P, Q P is a sufficient condition for QP is a sufficient condition for Q Q if PQ if P Q whenever PQ whenever P Q is a necessary condition for PQ is a necessary condition for P –Terminology: P = premise, hypothesis, antecedentP = premise, hypothesis, antecedent Q = conclusion, consequenceQ = conclusion, consequence Implication (if - then)

7. 7 Binary Operator, Symbol:  PQ PQPQPQPQ TTT TFF FTT FFT

8. 8 Biconditional (if and only if (iff)) Binary Operator, Symbol:  PQ PQPQPQPQ TTT TFF FTF FFT Both P and Q must have the same truth value.

9. Step 1: The first n columns of the table are labeled by the component propositional variables. Further columns are included for all intermediate combinations of the variables, culminating in a column for the full statement. Step 2: Under each of the first n headings, we list the 2 n possible n- tuples of truth values for the n component statements. Step 3: For each of the remaining columns, we compute, in sequence, the remaining truth values. Algorithm for making Truth Table

10. Statements and Operators Statements and operators can be combined in any way to form new statements. PQ PPPP QQQQ (  P)  (  Q) TTFFF TFFTT FTTFT FFTTT