Review Propositional Logic Propositions atomic and compound Operators: negation, and, or, xor, implies, biconditional Truth tables A closer look at implies Translating from/ to English Converse, inverse, contrapositive Logical Equivalence Tautology, contradiction, contingency Showing logical equivalence: truth table, or a proof Some key logical equivalences next pages
Key Logical Equivalences 1 Identity Laws: pù T º p, pú F º p Domination Laws: pú T º T, pù F º F Idempotent laws: pú pº p, pù p º p Double Negation Law: ( p) º p Negation Laws: pú pº T, pù pº F
Key Logical Equivalences 2 Commutative Laws: pú qº qú p, pù qº qù p Associative Laws: ( pùq) Ù r º pù( qùr) Distributive Laws: ( ) ( púq) Ú r º pú( qúr) ( pú qù r ) º ( púq) Ù( púr) ( pù( qú r) ) º ( pùq) Ú( pùr) Absorption Laws: pú( pù q) º p pù( pú q) º p
More Logical Equivalences TABLE 7 Logical Equivalences Involving Conditional Statements. p qº pú q p qº q p pú qº p q pù qº ( p q) ( p q) º pù q ( p q) Ù ( p r) º p ( qùr) ( p r) Ù ( q r) º ( pú q) r ( p q) Ú ( p r) º p ( qúr) ( p r) Ú ( q r) º ( pù q) r TABLE 8 Logical Equivalences Involving Biconditional Statements. ( ) ( ) p«qº p q Ù q p p«qº p«q «º ( Ù ) Ú ( Ù ) ( p q) p q p q p q p q «º «
( ) ( ) De Morgan s Laws pù q º pú q pú q º pù q This truth table shows that De Morgan s Second Law holds. p q p q (p q) (p q) p q T T F F T F F T F F T T F F F T T F T F F F F T T F T T
Constructing New Logical Equivalences We can show that two expressions are logically equivalent by developing a series of logically equivalent statements. To prove that A B we produce a series of equivalences beginning with A and ending with B. A º A 1 A n º B
Equivalence Proofs 1 ( ) Example: Show that pú ( pù q) Solution: is logically equivalent to pù q ( p ( p q) ) p ( p q) ë ( ) p ( p q) Ú Ù º Ù Ù by the second De Morgan law º pùé p Ú qù by the first De Morgan law û º Ù Ú by the double negation law º ( pù p) Ú ( pù q) by the second distributive law º FÚ ( pù q) because pù pº F º ( pù q) Ú F by the commutative law for disjunction º ( pù q) By the identity law for F
Equivalence Proofs 2 Example: Show that ( pù q) ( púq) is a tautology. Solution: pù q pú q º pùq Ú púq by truth table for ( ) ( ) ( ) ( ) º ( pú q) Ú( púq) by the first De Morgan law º ( pú p) Ú ( qú q) by associative and commutative laws laws for disjunction º T ÚT by truth tables º T by the domination law
Problem How many rows are there in a truth table with n propositional variables? 2 n, also written as 2^n. (We will see why later, when studying Counting.) With n propositional variables, how many distinct (that is, not equivalent) propositions can we have? 2^(2^n)
Problem Section 1.2 Applications of Propositional Logic not covered in class Read the sub-section Translating English Sentences on your own`
Propositional Satisfiability A compound proposition is satisfiable if there is an assignment of truth values to its variables that make it true. When no such assignments exist, the compound proposition is unsatisfiable. A compound proposition is unsatisfiable if and only if its negation is a tautology.
Propositional Satisfiability Example: Determine the satisfiability of the following: ( pú q) Ù( qú r) Ù( rú p) Solution: Satisfiable. Assign T to p, q, and r. ( púqúr) Ù ( pú qú r) Solution: Satisfiable. Assign T to p and F to q. ( pú q) Ù( qú r) Ù( rú p) Ù( púqúr) Ù ( pú qú r) Solution: Not satisfiable. Check each possible assignment of truth values to the propositional variables and none will make the proposition true.
First-Order Predicate Logic If we have: All men are mortal. and Socrates is a man. Does it follow that Socrates is mortal?
Predicate Logic A predicate is a function from some domain U to the range {T, F} Variables x, y, over the domain. Predicates P(x), Q(x,y), The statement P(x) is said to be the value of the predicate P at x. Note that for a given value of x, P(x) is a proposition. (Hence, predicates are also called propositional functions.) For example, let P(x) denote x > 0 and the domain be the integers. P(-3) is false. P(0) is false. P(3) is true.
Examples of Propositional Functions Let x + y = z be denoted by R(x, y, z) and U (for all three variables) be the integers. Find these truth values: R(2,-1,5) Solution: F R(3,4,7) Solution: T R(x, 3, z) Solution: Not a Proposition Now let x - y = z be denoted by Q(x, y, z), with U as the integers. Find these truth values: Q(2,-1,3) Solution: T Q(3,4,7) Solution: F Q(x, 3, z) Solution: Not a Proposition
Compound Expressions Connectives from propositional logic carry over to predicate logic. If P(x) denotes x > 0, find these truth values: P(3) P(-1) P(3) P(-1) P(3) P(-1) P(3) P(-1) Expressions with variables are not propositions and therefore do not have truth values. For example, P(3) P(y) P(x) P(y)
Quantifiers We need quantifiers to express the meaning of English words including all and some: All men are Mortal. Some cats do not have fur. The two most important quantifiers are: Universal Quantifier, For all, symbol: " Existential Quantifier, There exists, symbol: $ We write as in "x P(x) and $x P(x). "x P(x) asserts P(x) is true for every x in the domain. $x P(x) asserts P(x) is true for some x in the domain. The quantifiers are said to bind the variable x in these expressions.
Universal Quantifier "x P(x) is read as For all x, P(x) or For every x, P(x) Examples: 1) If P(x) denotes x > 0 and U is the integers, then "x P(x) is false. 2) If P(x) denotes x > 0 and U is the positive integers, then "x P(x) is true. 3) If P(x) denotes x is even and U is the integers, then " x P(x) is false.
Existential Quantifier $x P(x) is read as For some x, P(x), or as There is an x such that P(x), or For at least one x, P(x). Examples: 1. If P(x) denotes x > 0 and U is the integers, then $x P(x) is true. It is also true if U is the positive integers. 2. If P(x) denotes x < 0 and U is the positive integers, then $x P(x) is false. 3. If P(x) denotes x is even and U is the integers, then $x P(x) is true.
Thinking about Quantifiers When the domain of discourse is finite, we can think of quantification as looping through the elements of the domain. To evaluate "x P(x) loop through all x in the domain. If at every step P(x) is true, then "x P(x) is true. If at a step P(x) is false, then "x P(x) is false and the loop terminates. To evaluate $x P(x) loop through all x in the domain. If at some step, P(x) is true, then $x P(x) is true and the loop terminates. If the loop ends without finding an x for which P(x) is true, then $x P(x) is false. Even if the domains are infinite, we can still think of the quantifiers this fashion, but the loops will not terminate in some cases.