february 26, 2015applied discrete mathematics week 5: mathematical reasoning 1 addition of integers...
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February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Addition of Integers
Example:Add a = (1110)2 and b = (1011)2.
a0 + b0 = 0 + 1 = 02 + 1, so that c0 = 0 and s0 = 1.
a1 + b1 + c0 = 1 + 1 + 0 = 12 + 0, so c1 = 1 and s1 = 0.
a2 + b2 + c1 = 1 + 0 + 1 = 12 + 0, so c2 = 1 and s2 = 0.
a3 + b3 + c2 = 1 + 1 + 1 = 12 + 1, so c3 = 1 and s3 = 1.
s4 = c3 = 1.
Therefore, s = a + b = (11001)2.
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Addition of Integers
procedure add(a, b: positive integers)c := 0for j := 0 to n-1 {larger integer (a or b) has n digits}begin
d := (aj + bj + c)/2sj := aj + bj + c – 2dc := d
endsn := c{the binary expansion of the sum is (snsn-1…s1s0)2}
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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MatricesA matrix is a rectangular array of numbers.A matrix with m rows and n columns is called anmn matrix.
Example:
083.05.2
11A is a 32 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, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Matrices
A general description of an mn matrix A = [aij]:
mnmm
n
n
aaa
aaaaaa
A
............
...
...
21
22221
11211
inii aaa ...,,, 21
mj
j
j
a
aa
.
.
.2
1
i-th row of A
j-th column of A
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Matrix Addition
Let A = [aij] and B = [bij] be mn matrices.The sum of A and B, denoted by A+B, is the mnmatrix that has aij + bij as its (i, j)th element.In other words, A+B = [aij + bij].
Example:
17141103
146395
038412
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Matrix Multiplication
Let A be an mk matrix and B be a kn matrix.The product of A and B, denoted by AB, is the mnmatrix 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
tj
k
titkjikjijiij babababac
1
2211 ...
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Matrix Multiplication
A more intuitive description of calculating C = AB:
011500412103
A
4310
12B
- 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: 32 + 00 + 13 = 9 - Enter the result in the upper-left corner of C
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Matrix Multiplication
- Now superimpose the first column of B on the second, third, …, 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, …, 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.
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Matrix Multiplication
Let us calculate the complete matrix C:
011500412103
A
4310
12B
C98
15-2
71520-2
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Identity Matrices
The identity matrix of order n is the nn matrix In = [ij], where ij = 1 if i = j and ij = 0 if i j:
1...00.........0...100...01
A
Multiplying an mn matrix A by an identity matrix of appropriate size does not change this matrix:
AIn = ImA = A
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Powers and Transposes of Matrices
The power function can be defined for square matrices. If A is an nn matrix, we have:
A0 = In,Ar = AAA…A (r times the letter A)
The transpose of an mn matrix A = [aij], denoted by At, is the nm 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 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Powers and Transposes of Matrices
Example:
411302tA
4310
12A
A square matrix A is called symmetric if A = At.Thus A = [aij] is symmetric if aij = aji for alli = 1, 2, …, n and j = 1, 2, …, n.
493921
315A
131131131
B
A is symmetric, B is not.
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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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 ab
0 0 0
0 1 0
1 0 0
1 1 1
a b ab
0 0 0
0 1 1
1 0 1
1 1 1
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Zero-One Matrices
Let A = [aij] and B = [bij] be mn 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, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Zero-One Matrices
Example:
011011
A
001110
B
Join:
011111
000111101101
BA
Meet:
001010
000111101101
BA
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Zero-One MatricesLet A = [aij] be an mk zero-one matrix and B = [bij] be a kn zero-one matrix.
Then the Boolean product of A and B, denoted by AB, is the mn 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 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Zero-One Matrices
Example:
11
01A
10
10B
10
10)11()11()01()01()10()11()00()01(
BA
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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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] = AA…A (r times the letter A)
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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Let’s proceed to…
Mathematical Reasoning
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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 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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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, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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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 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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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(pq)) q. We write it in the following way:
pp q____ q
The two hypotheses p and p q are written in a column, and the conclusionbelow a bar, where means “therefore”.
February 26, 2015 Applied Discrete Mathematics Week 5: Mathematical Reasoning
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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.