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February 26, 2013 Applied Discrete Mathematics Week 5: Mathematical Reasoning 1
Matrices
A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m ×××× n matrix.
Example: is a 3×2 matrix.
A matrix with the same number of rows and columns is called square.
Two matrices are equal if they have the same number of rows and columns and the corresponding entries in every position are equal.
February 26, 2013 Applied Discrete Mathematics Week 5: Mathematical Reasoning 2
Matrices
A general description of an m×n matrix A = [aij]:
i-th row of A
j-th column of A
February 26, 2013 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Matrix Addition
Let A = [aij] and B = [bij] be m×n matrices. The sum of A and B, denoted by A+B, is the m×n matrix that has aij + bij as its (i, j)th element. In other words, A+B = [aij + bij].
Example:
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Matrix Multiplication
Let A be an m×k matrix and B be a k×n matrix. The product of A and B, denoted by AB, is the m×n matrix with (i, j)th entry equal to the sum of the products of the corresponding elements from the i-th row of A and the j-th column of B.
In other words, if AB = [cij], then
February 26, 2013 Applied Discrete Mathematics Week 5: Mathematical Reasoning 5
Matrix Multiplication
A more intuitive description of calculating C = AB:
- Take the first column of B
- Turn it counterclockwise by 90° and superimpose it on the first row of A
- Multiply corresponding entries in A and B and add the products: 3⋅2 + 0⋅0 + 1⋅3 = 9
- Enter the result in the upper-left corner of C February 26, 2013 Applied Discrete Mathematics Week 5: Mathematical Reasoning 6
Matrix Multiplication
- Now superimpose the first column of B on the second, third, 5, m-th row of A to obtain the entries in the first column of C (same order).
- Then repeat this procedure with the second, third, 5, n-th column of B, to obtain to obtain the remaining columns in C (same order).
- After completing this algorithm, the new matrix C contains the product AB.
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Matrix Multiplication
Let us calculate the complete matrix C:
February 26, 2013 Applied Discrete Mathematics Week 5: Mathematical Reasoning 8
Identity 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:
Multiplying an m×n matrix A by an identity matrix of appropriate size does not change this matrix: AIn = ImA = A
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Powers and Transposes of Matrices
The power function can be defined for square matrices. If A is an n×n matrix, we have:
A^0 = In, Ar^ = AAA5A (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, 5, n and j = 1, 2, 5, m.
February 26, 2013 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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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, 5, n and j = 1, 2, 5, n.
A is symmetric, B is not.
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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 0 0 0 0 1 0 1 0 0 1 1 1
a b a∨b 0 0 0 0 1 1 1 0 1 1 1 1 February 26, 2013 Applied Discrete Mathematics Week 5: Mathematical Reasoning 12
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 26, 2013 Applied Discrete Mathematics Week 5: Mathematical Reasoning 19
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 26, 2013 Applied Discrete Mathematics Week 5: Mathematical Reasoning 20
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.
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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”.
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Rules of Inference
The general form of a rule of inference is:
p 1 p 2 . . . pn
∴ q
The rule states that if p 1 and p 2 and 5 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.
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Rules of Inference
p
∴ p∨q Addition
p∧q
∴ p Simplification
p q
∴ p∧q
Conjunction
¬q p→q
∴ ¬p
Modus tollens
p→q q→r
∴ p→r
Hypothetical syllogism
p∨q ¬p
∴ q
Disjunctive syllogism
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Arguments
Just like a rule of inference, an argument consists of one or more hypotheses and a conclusion.
We say that an argument is valid , if whenever all its hypotheses are true, its conclusion is also true.
However, if any hypothesis is false, even a valid argument can lead to an incorrect conclusion.
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Arguments
Example: “If 101 is divisible by 3, then 101^2 is divisible by 9. 101 is divisible by 3. Consequently, 101^2 is divisible by 9.”
Although the argument is valid , its conclusion is incorrect , because one of the hypotheses is false (“101 is divisible by 3.”).
If in the above argument we replace 101 with 102, we could correctly conclude that 102^2 is divisible by
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Arguments
Which rule of inference was used in the last argument?
p: “101 is divisible by 3.” q: “101^2 is divisible by 9.” p p→q
∴ q
Modus ponens
Unfortunately, one of the hypotheses (p) is false. Therefore, the conclusion q is incorrect.
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Arguments
Another example:
“If it rains today, then we will not have a barbeque today. If we do not have a barbeque today, then we will have a barbeque tomorrow. Therefore, if it rains today, then we will have a barbeque tomorrow.”
This is a valid argument: If its hypotheses are true, then its conclusion is also true.
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Arguments
Let us formalize the previous argument:
p: “It is raining today.” q: “We will not have a barbecue today.” r: “We will have a barbecue tomorrow.”
So the argument is of the following form:
p→q q→r
∴ p→r
Hypothetical syllogism
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Arguments
Another example:
Gary is either intelligent or a good actor. If Gary is intelligent, then he can count from 1 to 10. Gary can only count from 1 to 2. Therefore, Gary is a good actor.
i: “Gary is intelligent.” a: “Gary is a good actor.” c: “Gary can count from 1 to 10.”
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Arguments
i: “Gary is intelligent.” a: “Gary is a good actor.” c: “Gary can count from 1 to 10.”
Step 1: ¬c Hypothesis Step 2: i → c Hypothesis Step 3: ¬i Modus Tollens Steps 1 & 2 Step 4: a ∨ i Hypothesis Step 5: a Disjunctive Syllogism Steps 3 & 4
Conclusion: a (“Gary is a good actor.”)
