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(. ) 2. Addition of Integers. carry. 1. 1. 1. How do we (humans) add two integers? Example: 7583 +4932. 1. 2. 5. 1. 5. carry. 1. 1. Binary expansions: (1011) 2 + (1010) 2. 1. 0. 1. 0. 1. - PowerPoint PPT Presentation
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February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Addition of IntegersAddition of IntegersHow do we (humans) add two integers?How do we (humans) add two integers?
Example: Example: 75837583 + + 49324932
5511552211
111111 carrycarry
Binary expansions: Binary expansions: (1011)(1011)22 + + (1010)(1010)22
1100
carrycarry11
1100
11
11(( ))22
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Addition of IntegersAddition of IntegersLet a = (aLet a = (an-1n-1aan-2n-2…a…a11aa00))22, b = (b, b = (bn-1n-1bbn-2n-2…b…b11bb00))2.2.
How can we How can we algorithmically algorithmically add these two binary add these two binary numbers?numbers?First, add their rightmost bits:First, add their rightmost bits:aa00 + b + b00 = c = c002 + s2 + s00,,where swhere s00 is the is the rightmost bitrightmost bit in the binary in the binary expansion of a + b, and cexpansion of a + b, and c00 is the is the carrycarry..Then, add the next pair of bits and the carry:Then, add the next pair of bits and the carry:aa11 + b + b1 1 + c+ c00 = c = c112 + s2 + s11,,where swhere s11 is the is the next bitnext bit in the binary expansion of in the binary expansion of a + b, and ca + b, and c11 is the carry. is the carry.
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Addition of IntegersAddition of Integers
Continue this process until you obtain cContinue this process until you obtain cn-1n-1..
The leading bit of the sum is sThe leading bit of the sum is snn = c = cn-1n-1..
The result is:The result is:a + b = (sa + b = (snnssn-1n-1…s…s11ss00))22
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Addition of IntegersAddition of IntegersExample:Example:Add a = (1110)Add a = (1110)22 and b = (1011) and b = (1011)22..
aa00 + b + b00 = 0 + 1 = 0 = 0 + 1 = 02 + 1, so that c2 + 1, so that c00 = 0 and s = 0 and s00 = 1. = 1.aa11 + b + b1 1 + c+ c00 = 1 + 1 + 0 = 1 = 1 + 1 + 0 = 12 + 0, so c2 + 0, so c11 = 1 and s = 1 and s11 = 0. = 0.aa22 + b + b2 2 + c+ c11 = 1 + 0 + 1 = 1 = 1 + 0 + 1 = 12 + 0, so c2 + 0, so c22 = 1 and s = 1 and s22 = 0. = 0.aa33 + b + b3 3 + c+ c22 = 1 + 1 + 1 = 1 = 1 + 1 + 1 = 12 + 1, so c2 + 1, so c33 = 1 and s = 1 and s33 = 1. = 1.ss44 = c = c33 = 1. = 1.
Therefore, s = a + b = (11001)Therefore, s = a + b = (11001)22..
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Addition of IntegersAddition of Integersprocedure procedure add(a, b: positive integers)add(a, b: positive integers)c := 0c := 0for j := 0 to n-1for j := 0 to n-1beginbegin
d := d := (a(ajj + b + bjj + c)/2 + c)/2ssjj := a := ajj + b + bjj + c – 2d + c – 2dc := dc := d
endendssnn := c := c{the binary expansion of the sum is (s{the binary expansion of the sum is (snnssn-1n-1……ss11ss00))22}}
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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MatricesMatricesA A matrixmatrix is a rectangular array of numbers. is a rectangular array of numbers.A matrix with m rows and n columns is called A matrix with m rows and n columns is called ananmmn matrix.n matrix.
Example:Example:
083.05.2
11A is a 3is a 32 matrix.2 matrix.
A matrix with the same number of rows and A matrix with the same number of rows and columns is called columns is called squaresquare..Two matrices are Two matrices are equalequal if they have the same if they have the same number of rows and columns and the number of rows and columns and the corresponding entries in every position are corresponding entries in every position are equal. equal.
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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MatricesMatricesA general description of an mA general description of an mn matrix A = [an matrix A = [aijij]: ]:
mnmm
n
n
aaa
aaaaaa
A
............
...
...
21
22221
11211
inii aaa ...,,, 21
mj
j
j
a
aa
.
.
.2
1
i-th row of Ai-th row of A
j-th column j-th column
of A of A
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Matrix AdditionMatrix AdditionLet A = [aLet A = [aijij] and B = [b] and B = [bijij] be m] be mn matrices.n matrices.The sum of A and B, denoted by A+B, is the mThe sum of A and B, denoted by A+B, is the mnnmatrix that has amatrix that has aijij + b + bijij as its (i, j)th element. as its (i, j)th element.In other words, A+B = [aIn other words, A+B = [aijij + b + bijij]. ].
Example:Example:
17141103
146395
038412
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Matrix MultiplicationMatrix MultiplicationLet A be an mLet A be an mk matrix and B be a kk matrix and B be a kn matrix.n matrix.The The productproduct of A and B, denoted by AB, is the of A and B, denoted by AB, is the mmnnmatrix with (i, j)th entry equal to the sum of the matrix with (i, j)th entry equal to the sum of the products of the corresponding elements from products of the corresponding elements from the i-th row of A and the j-th column of B.the i-th row of A and the j-th column of B.
In other words, if AB = [cIn other words, if AB = [cijij], then], thentj
k
titkjikjijiij babababac
1
2211 ...
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Matrix MultiplicationMatrix MultiplicationA more intuitive description of calculating C = AB: A more intuitive description of calculating C = AB:
011500412103
A
4310
12B
- Take the first column of B - Take the first column of B - Turn it counterclockwise by 90Turn it counterclockwise by 90 and and superimpose superimpose it on the first row of A it on the first row of A - Multiply corresponding entries in A and B andMultiply corresponding entries in A and B and add the products: 3 add the products: 32 + 02 + 00 + 10 + 13 = 9 3 = 9 - Enter the result in the upper-left corner of C Enter the result in the upper-left corner of C
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Matrix MultiplicationMatrix Multiplication- Now superimpose the first column of B on the Now superimpose the first column of B on the second, third, …, m-th row of A to obtain the second, third, …, m-th row of A to obtain the entries in the first column of C (same order). entries in the first column of C (same order).
- Then repeat this procedure with the second, Then repeat this procedure with the second, third, …, n-th column of B, to obtain to obtain third, …, n-th column of B, to obtain to obtain the remaining columns in C (same order). the remaining columns in C (same order).
- After completing this algorithm, the new matrixAfter completing this algorithm, the new matrix C contains the product AB. C contains the product AB.
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Matrix MultiplicationMatrix MultiplicationLet us calculate the complete matrix C: Let us calculate the complete matrix C:
011500412103
A
4310
12B
C9988
1515-2-2
7715152020-2-2
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Identity MatricesIdentity MatricesThe The identity matrix of order nidentity matrix of order n is the n is the nn matrix n matrix IInn = [ = [ijij], where ], where ij ij = 1 if i = j and = 1 if i = j and ij ij = 0 if i = 0 if i j: j:
1...00.........0...100...01
A
Multiplying an mMultiplying an mn matrix A by an identity n matrix A by an identity matrix of appropriate size does not change this matrix of appropriate size does not change this matrix:matrix:AIAInn = I = ImmA = AA = A
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Powers and Transposes of Powers and Transposes of MatricesMatrices
The The power functionpower function can be defined for can be defined for squaresquare matrices. If A is an nmatrices. If A is an nn matrix, we have:n matrix, we have:
AA00 = I = Inn,,AArr = AAA…A = AAA…A (r times the letter A)(r times the letter A)
The The transpose transpose of an mof an mn matrix A = [an matrix A = [aijij], ], denoted by Adenoted by Att, is the n, is the nm matrix obtained by m matrix obtained by interchanging the rows and columns of A.interchanging the rows and columns of A.
In other words, if AIn other words, if Att = [b = [bijij], the b], the bijij = a = ajiji for for i = 1, 2, …, n and j = 1, 2, …, m.i = 1, 2, …, n and j = 1, 2, …, m.
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Powers and Transposes of Powers and Transposes of MatricesMatrices
Example:Example:
411302tA
4310
12A
A square matrix A is called A square matrix A is called symmetricsymmetric if A = if A = AAtt..Thus A = [aThus A = [aijij] is symmetric if a] is symmetric if aij ij = a= ajiji for all for alli = 1, 2, …, n and j = 1, 2, …, m.i = 1, 2, …, n and j = 1, 2, …, m.
493921
315A
131131131
B
A is symmetric, B is not.A is symmetric, B is not.
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Zero-One MatricesZero-One MatricesA matrix with entries that are either 0 or 1 is A matrix with entries that are either 0 or 1 is called a called a zero-one matrixzero-one matrix. Zero-one matrices . Zero-one matrices are often used like a “table” to represent are often used like a “table” to represent discrete structures.discrete structures.We can define Boolean operations on the We can define Boolean operations on the entries in zero-one matrices:entries in zero-one matrices:
aa bb aabb00 00 0000 11 0011 00 0011 11 11
aa bb aabb00 00 0000 11 1111 00 1111 11 11
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Zero-One MatricesZero-One MatricesLet A = [aLet A = [aijij] and B = [b] and B = [bijij] be m] be mn zero-one n zero-one matrices.matrices.
Then the Then the joinjoin of A and B is the zero-one matrix of A and B is the zero-one matrix with (i, j)th entry awith (i, j)th entry aijij b bijij. The join of A and B is . The join of A and B is denoted by A denoted by A B. B.
The The meetmeet of A and B is the zero-one matrix of A and B is the zero-one matrix with (i, j)th entry awith (i, j)th entry aijij b bijij. The meet of A and B is . The meet of A and B is denoted by A denoted by A B. B.
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Zero-One MatricesZero-One Matrices
Example:Example:
011011
A
001110
B
Join:Join:
011111
000111101101
BA
Meet:Meet:
001010
000111101101
BA
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Zero-One MatricesZero-One MatricesLet A = [aLet A = [aijij] be an m] be an mk zero-one matrix and k zero-one matrix and B = [bB = [bijij] be a k] be a kn zero-one matrix.n zero-one matrix.Then the Then the Boolean productBoolean product of A and B, of A and B, denoted by Adenoted by AB, is the mB, is the mn matrix with (i, j)th n matrix with (i, j)th entry [centry [cijij], where], wherecij = (acij = (ai1i1 b b1j1j) ) (a (ai2i2 b b2i2i) ) … … (a (aikik b bkjkj). ). Note that the actual Boolean product symbol Note that the actual Boolean product symbol has a dot in its center.has a dot in its center.Basically, Boolean multiplication works like the Basically, Boolean multiplication works like the multiplication of matrices, but with computing multiplication of matrices, but with computing instead of the product and instead of the product and instead of the sum. instead of the sum.
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Zero-One MatricesZero-One Matrices
Example:Example:
11
01A
10
10B
10
10)11()11()01()01()10()11()00()01(BA
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Zero-One MatricesZero-One MatricesLet A be a square zero-one matrix and r be a Let A be a square zero-one matrix and r be a positive integer.positive integer.
The The r-th Boolean powerr-th Boolean power of A is the Boolean of A is the Boolean product of r factors of A. The r-th Boolean product of r factors of A. The r-th Boolean power of A is denoted by Apower of A is denoted by A[r][r]..
AA[0][0] = I = Inn,,AA[r][r] = A = AAA……A A (r times the letter A)(r times the letter A)
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Let’s proceed to…Let’s proceed to…
Mathematical Mathematical ReasoningReasoning
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Mathematical ReasoningMathematical Reasoning
We need We need mathematical reasoningmathematical reasoning to to• determine whether a mathematical determine whether a mathematical argument is argument is correct or incorrect and correct or incorrect and• construct mathematical arguments.construct mathematical arguments.
Mathematical reasoning is not only important Mathematical reasoning is not only important for conducting for conducting proofsproofs and and program program verificationverification, but also for , but also for artificial artificial intelligenceintelligence systems (drawing inferences). systems (drawing inferences).
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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TerminologyTerminologyAn An axiomaxiom is a basic assumption about is a basic assumption about mathematical structured that needs no proof.mathematical structured that needs no proof.We can use a We can use a proofproof to demonstrate that a to demonstrate that a particular statement is true. A proof consists of a particular statement is true. A proof consists of a sequence of statements that form an argument.sequence of statements that form an argument.The steps that connect the statements in such a The steps that connect the statements in such a sequence are the sequence are the rules of inferencerules of inference..Cases of incorrect reasoning are called Cases of incorrect reasoning are called fallaciesfallacies..A A theoremtheorem is a statement that can be shown to is a statement that can be shown to be true. be true.
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TerminologyTerminology
A A lemmalemma is a simple theorem used as an is a simple theorem used as an intermediate result in the proof of another intermediate result in the proof of another theorem.theorem.
A A corollarycorollary is a proposition that follows is a proposition that follows directly from a theorem that has been proved.directly from a theorem that has been proved.
A A conjectureconjecture is a statement whose truth is a statement whose truth value is unknown. Once it is proven, it value is unknown. Once it is proven, it becomes a theorem.becomes a theorem.
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Rules of InferenceRules of InferenceRules of inferenceRules of inference provide the justification of provide the justification of the steps used in a proof.the steps used in a proof.
One important rule is called One important rule is called modus ponensmodus ponens or the or the law of detachmentlaw of detachment. It is based on the . It is based on the tautology tautology (p(p(p(pq)) q)) q. We write it in the following way: q. We write it in the following way:
ppp p q q________ qq
The two The two hypotheseshypotheses p and p p and p q are q are
written in a column, and the written in a column, and the conclusionconclusionbelow a bar, where below a bar, where means means “therefore”.“therefore”.
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Rules of InferenceRules of Inference
The general form of a rule of inference is:The general form of a rule of inference is:
pp11 pp22 .. .. .. ppnn________ qq
The rule states that if pThe rule states that if p11 andand p p22 andand … … andand p pnn are all true, then q is true as are all true, then q is true as well.well.
These rules of inference can be used These rules of inference can be used in any mathematical argument and do in any mathematical argument and do not not require any proof.require any proof.
February 25, 2002 Applied Discrete MathematicsWeek 5: Mathematical Reasoning
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Rules of InferenceRules of Inference pp__________ ppqq AdditionAddition
ppqq__________ pp SimplificatioSimplificatio
nn pp qq__________ ppqq
ConjunctionConjunction
qq ppq q __________ pp
Modus Modus tollenstollens
ppqq qqr r __________ ppr r
Hypothetical Hypothetical syllogismsyllogism
ppqq pp__________ q q
Disjunctive Disjunctive syllogismsyllogism