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)

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

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)

Set operations (union, intersection, difference, complement) 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)

Functions (section 1.8) Basics Types of functions 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

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

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

Types of Questions Conceptual Problem solving Proofs 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