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2/21/2017 Identity Matrices Powers and Transposes of Matrices The identity matrix of order n is the n×n matrix In = [δij], where δij = 1 if i = j and δij = 0 if i ≠ j: The power function can be defined for square matrices. If A is an n×n matrix, we have: A0 = In, Ar = AAA…A (r times the letter A) The transpose of an m×n matrix A = [aij], denoted by At, is the n×m matrix obtained by interchanging the rows and columns of A. Multiplying an m×n matrix A by an identity matrix of appropriate size does not change this matrix: AIn = ImA = A February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning 1 In other words, if At = [bij], then bij = aji for i = 1, 2, …, n and j = 1, 2, …, m. February 21, 2017 A matrix with entries that are either 0 or 1 is called a zero-one matrix. Zero-one matrices are often used like a “table” to represent discrete structures. Example: We can define Boolean operations on the entries in zero-one matrices: A square matrix A is called symmetric if A = At. Thus A = [aij] is symmetric if aij = aji for all i = 1, 2, …, n and j = 1, 2, …, n. A is symmetric, B is not. Applied Discrete Mathematics Week 5: Mathematical Reasoning 3 a b a∧b a b a∨b 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 1 1 1 February 21, 2017 Zero-One Matrices Then the join of A and B is the zero-one matrix with (i, j)th entry aij ∨ bij. The join of A and B is denoted by A ∨ B. The meet of A and B is the zero-one matrix with (i, j)th entry aij ∧ bij. The meet of A and B is denoted by A ∧ B. Applied Discrete Mathematics Week 5: Mathematical Reasoning Applied Discrete Mathematics Week 5: Mathematical Reasoning 4 Zero-One Matrices Let A = [aij] and B = [bij] be m×n zero-one matrices. February 21, 2017 2 Zero-One Matrices Powers and Transposes of Matrices February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning 5 Example: Join: Meet: February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning 6 1 2/21/2017 Zero-One Matrices Zero-One Matrices Let A = [aij] be an m×k zero-one matrix and B = [bij] be a k×n zero-one matrix. Then the Boolean product of A and B, denoted by AοB, is the m×n matrix with (i, j)th entry [cij], where Example: cij = (ai1 ∧ b1j) ∨ (ai2 ∧ b2i) ∨ … ∨ (aik ∧ bkj). Note that the actual Boolean product symbol has a dot in its center. Basically, Boolean multiplication works like the multiplication of matrices, but with computing ∧ instead of the product and ∨ instead of the sum. February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning 7 February 21, 2017 Zero-One Matrices Applied Discrete Mathematics Week 5: Mathematical Reasoning 8 Let’s proceed to… Let A be a square zero-one matrix and r be a positive integer. The r-th Boolean power of A is the Boolean product of r factors of A. The r-th Boolean power of A is denoted by A[r]. A[0] = In, A[r] = AοAο…οA February 21, 2017 Mathematical Reasoning (r times the letter A) Applied Discrete Mathematics Week 5: Mathematical Reasoning 9 February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning 10 Terminology Mathematical Reasoning We need mathematical reasoning to An axiom is a basic assumption about mathematical structures that needs no proof. • determine whether a mathematical argument is correct or incorrect and • construct mathematical arguments. We can use a proof to demonstrate that a particular statement is true. A proof consists of a sequence of statements that form an argument. Mathematical reasoning is not only important for conducting proofs and program verification, but also for artificial intelligence systems (drawing inferences). The steps that connect the statements in such a sequence are the rules of inference. February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning 11 Cases of incorrect reasoning are called fallacies. A theorem is a statement that can be shown to be true. February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning 12 2 2/21/2017 Terminology Rules of Inference A lemma is a simple theorem used as an intermediate result in the proof of another theorem. A corollary is a proposition that follows directly from a theorem that has been proved. A conjecture is a statement whose truth value is unknown. Once it is proven, it becomes a theorem. February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning 13 Rules of inference provide the justification of the steps used in a proof. One important rule is called modus ponens or the law of detachment. It is based on the tautology (p∧(p→q)) → q. We write it in the following way: p p→q ____ ∴q The two hypotheses p and p → q are written in a column, and the conclusion below a bar, where ∴ means “therefore”. February 21, 2017 Rules of Inference The rule states that if p1 and p2 and … and pn are all true, then q is true as well. These rules of inference can be used in any mathematical argument and do not require any proof. February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning 14 Rules of Inference The general form of a rule of inference is: p1 p2 . . . pn ____ ∴q Applied Discrete Mathematics Week 5: Mathematical Reasoning 15 p _____ ∴ p∨q Addition ¬q p→q _____ ∴ ¬p p∧q _____ ∴p Simplification p→q q→r _____ ∴ p→r Hypothetical syllogism p q _____ ∴ p∧q Conjunction p∨q ¬p _____ ∴q Disjunctive syllogism February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning Modus tollens 16 3