A Brief Summary for Exam 1

1. A Brief Summary for Exam 1 Subject Topics • Propositional Logic (sections 1.1, 1.2) • Propositions • Statement, Truth value, • Proposition, Propositional symbol, Open proposition • Operators (, , , , ) • Define by truth tables • Composite propositions • Tautology and contradiction • Equivalence of propositional statements • Definition • Equivalence laws • Proving equivalence (by truth table or equivalence laws)

2. Predicate Logic (sections 1.3, 1.4) • Predicates • Universal and existential quantifiers, and the duality of the two (wrt negation) • When predicates have truth values (become propositions) • All of its variables are instantiated • All of its variables are quantified • Nested quantifiers • Quantifiers with negation • Logical expressions formed by predicates, operators, and quantifiers

3. Mathematical reasoning (proofs) (section 1.5) • Rules of inference • MP, MT, chaining, resolution, simplification, addition, etc. • Universal/ existential instantiation/generalization • Valid argument (hypotheses and conclusion) • Construction of valid argument using rules of inference • Write down each rule used, together with the statements used by the rule • Proof methods (proof if P then Q) • Direct proof: show if P true then Q must be true (i.e., P  Q  T) • Indirect proof: show that if Q is false then P must be false (its contrapositive is a tautology) • Prove by contradiction: assume Q is false then derive a contradiction (i.e., derive both r and  r for some r)

4. Set Theory (sections 1.6, 1.7) • Basics • Membership, subsets, cardinality, set equality • Defining sets: enumeration, builder function • Cartesian product • Power set • Set operations (union, intersection, difference, complement) • Definitions (in words and in logical expressions) • Set identity laws • Show two sets are equal (by identity laws and by membership table)

5. Functions (section 1.8) • Basics • What is a function (what are not function) • Domain, co-domain, range, image, pre-image • Types of functions • Injective (one-to-one), surjective (onto), bijective (one-to-one correspondence) • Inverse function • Composition of function

6. Boolean Algebra (sections 10.1, 10.2) • Boolean function and Boolean expression • Domain ({0,1}), Boolean variables • Boolean operations (sum, product, complement) • Define Boolean function by table • Two Boolean functions are equal if they have the same table • Minterms: generate Boolean expression from table • Correspondence between • Propositional logic • Sets • Boolean algebra

7. Algorithms (sections 2.1 – 2.3) • Algorithm and its properties • Definiteness, finiteness, and correctness • Complexity of algorithm • How much resource it takes to solve a problem • What’s important is the growth (maters only with large problems) • Big-O notation (upper bound) • Common growth functions • Useful rules for Big-O

8. Types of Questions • Conceptual • Definitions of terms • True/false • Multiple choice • Problem solving • Work with small concrete example problems • Proofs • Simple theorems or propositions • No questions will be outside of this summary and lecture notes