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AE4M33RZN Fuzzy logicTutorial examples

Radomiacuter Černochradomircernochfelcvutcz

Faculty of Electrical Engineering CTU in Prague

2012

Task 2

AssignmentOn the universeΔ = a b c d there is a fuzzy set

120583A = 1114106(a 03) (b 1) (c 05)1114109

Find its horizontal representation

120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

X 120572 = 0a b c 120572 isin (0 03⟩b c 120572 isin (03 05⟩b 120572 isin (05 1⟩

Task 3

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩

Find the vertical representation

Solution

120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore

120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109

Task 5

AssignmentOn the universeΔ = IR there is a fuzzy set A

120583A(x) =

⎧⎪⎪⎨⎪⎪⎩

x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise

Find its horizontal represenation

Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7


120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0

⟨1205722 1) otherwise


120583A(x) =⎧⎪⎨⎪⎩

radicx x isin (0 1)0 otherwise

Task 8

AssignmentDecide if the following function is a fuzzy conjunction

120572and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩

120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise

Solution

The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from

120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise

We get the same for 1114102120572and∘ 1205731114105and∘ 120574

It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)

Task 10


120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 11

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and

1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or

2 algebraic1113682or

3 Łukasiewicz 1113693or

Task 13

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 14

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard1113700or



Task 15

AssignmentDecide which equalities hold

1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)

Justify your conclusions

Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17

AssignmentVerify that 120572and∘ (120572

1113699rArr∘ 120573) = 120572and1113700120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0


AE4M33RZN Fuzzy logicTutorial examples

Radomiacuter Černochradomircernochfelcvutcz

Faculty of Electrical Engineering CTU in Prague

2012

Task 2


120583A = 1114106(a 03) (b 1) (c 05)1114109


120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩


Task 3


120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩



Solution


120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109

Task 5


120583A(x) =

⎧⎪⎪⎨⎪⎪⎩



Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7


120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0



120583A(x) =⎧⎪⎨⎪⎩


Task 8


120572and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩


Solution





Task 10



Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 2


120583A = 1114106(a 03) (b 1) (c 05)1114109


120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩


Task 3


120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩



Solution


120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109

Task 5


120583A(x) =

⎧⎪⎪⎨⎪⎪⎩



Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7


120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0



120583A(x) =⎧⎪⎨⎪⎩


Task 8


120572and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩


Solution





Task 10



Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 3


120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩



Solution


120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109

Task 5


120583A(x) =

⎧⎪⎪⎨⎪⎪⎩



Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7


120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0



120583A(x) =⎧⎪⎨⎪⎩


Task 8


120572and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩


Solution





Task 10



Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Solution


120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109

Task 5


120583A(x) =

⎧⎪⎪⎨⎪⎪⎩



Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7


120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0



120583A(x) =⎧⎪⎨⎪⎩


Task 8


120572and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩


Solution





Task 10



Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 5


120583A(x) =

⎧⎪⎪⎨⎪⎪⎩



Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7


120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0



120583A(x) =⎧⎪⎨⎪⎩


Task 8


120572and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩


Solution





Task 10



Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7


120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0



120583A(x) =⎧⎪⎨⎪⎩


Task 8


120572and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩


Solution





Task 10



Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 7


120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0



120583A(x) =⎧⎪⎨⎪⎩


Task 8


120572and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩


Solution





Task 10



Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 8


120572and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩


Solution





Task 10



Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Solution





Task 10



Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 10



Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 11


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 12


1113693(120572 ∘ornot

1113700120573) = 120572 where the

disjunction∘or is

1 standard1113700or



Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 13


2 rarr ⟨0 1⟩

120572and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise


Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 14


1 standard1113700or



Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 15


1 (120572and1113700120572) 1113693or(120572and

1113700120573) = 120572and

1113700(120572 1113693or120573)

2 (120572and1113693120572) 1113700or(120572and

1113693120573) = 120572and

1113693(120572 1113700or120573)

3 120572 1113700or(120572and1113693120573) = 120572and

1113693(120572 1113700or120573)


Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 16


1 (120572and1113700120573) and

1113693120574 = 120572and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or120573) = not

1113700120572and

1113693not1113700120573

3 (120572and1113693120572) 1113693ornot

1113700120572 = (not

1113700120572and

1113693not1113700120572) 1113693or120572


Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 17



1 algebraic ops

2 standard ops

Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Solution


120572and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572and

1113700120573

Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Task 19


R a b c d

ab 05c 03d 02

Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




Solution


R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1




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