Applied Discrete Mathematics Week 5: Mathematical Reasoning Identity Matrices The identity matrix of order n is the nn matrix In = [ij], where ij = 1 if i = j and ij = 0 if i  j: 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

Powers and Transposes of Matrices 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. In other words, if At = [bij], then bij = aji for i = 1, 2, …, n and j = 1, 2, …, m. February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning

Powers and Transposes of Matrices Example: 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. February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning

Applied Discrete Mathematics Week 5: Mathematical Reasoning Zero-One Matrices 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. We can define Boolean operations on the entries in zero-one matrices: a b ab 1 a b ab 1 February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning

Applied Discrete Mathematics Week 5: Mathematical Reasoning Zero-One Matrices Let A = [aij] and B = [bij] be mn 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. February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning

Applied Discrete Mathematics Week 5: Mathematical Reasoning Zero-One Matrices Example: Join: Meet: February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning

Applied Discrete Mathematics Week 5: Mathematical Reasoning 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 AB, is the mn matrix with (i, j)th entry [cij], where 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

Applied Discrete Mathematics Week 5: Mathematical Reasoning Zero-One Matrices Example: February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning

Applied Discrete Mathematics Week 5: Mathematical Reasoning Zero-One Matrices 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] = AA…A (r times the letter A) February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning

Mathematical Reasoning Let’s proceed to… Mathematical Reasoning February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning

Mathematical Reasoning We need mathematical reasoning to determine whether a mathematical argument is correct or incorrect and construct mathematical arguments. Mathematical reasoning is not only important for conducting proofs and program verification, but also for artificial intelligence systems (drawing inferences). February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning

Applied Discrete Mathematics Week 5: Mathematical Reasoning Terminology An axiom is a basic assumption about mathematical structures that needs no proof. 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. The steps that connect the statements in such a sequence are the rules of inference. 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

Applied Discrete Mathematics Week 5: Mathematical Reasoning Terminology 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

Applied Discrete Mathematics Week 5: Mathematical Reasoning Rules of Inference 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 Applied Discrete Mathematics Week 5: Mathematical Reasoning

Applied Discrete Mathematics Week 5: Mathematical Reasoning Rules of Inference The general form of a rule of inference is: p1 p2 . pn ____  q 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

Applied Discrete Mathematics Week 5: Mathematical Reasoning Rules of Inference q pq _____  p p _____  pq Modus tollens Addition pq qr _____  pr pq _____  p Hypothetical syllogism Simplification p q _____  pq pq p _____  q Conjunction Disjunctive syllogism February 21, 2017 Applied Discrete Mathematics Week 5: Mathematical Reasoning