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permutations as a bijection (different elements should be mapped to different elements, and every element needs to be mapped to an element) from ${1,2,3,4}$ to ${1,2,3,4}$: +$ mat(1,2,3,4;2,3,1,4) $ +Notice the 4, mapped unto itself, it's called a fixed point. The 4's would also be a fixed point here: +$ mat(1,2,4,3;2,3,4,1) $ +But then its a different set at the top. + +== Derangement +A bijection ($pi$) of a set `S` unto itself with no fixed points. + +That is $pi(i) eq.not i$ + +The hat problem: to find the chances of noone getting their own hat, just use derangements. + +== Counting permutations +$A$ is the set of all permutations of ${1,2,dots,n}$ + +$A_i =$ the set of permutations $pi$ such that $pi(i)=i$ + +$ +D_n &=|A\\(union_(i=1)^n) = |A| - sum_(i=1)^n + sum_(i_1Ysid@YgMc(hcd4KtsdS?t4IEe0# zv;Ixt3HDZJn#Ju?D#!ow zJv9_N2NMG)2l^TP)g1WVRpj?<^=%9co}&zepU^YoN~$NJQ1?*c!jBZ4;#NkTH5Ipy z1=jI+UR`8(m_%thefrHMJ&EVx66YD{pVKTB))#6%E@$X3j2&H6RyQ<^aV;^JS5m0O zVh^`!u~K0R7ZJWLe0KKeV2XB*(`OLV8W(MQe!RU=pClM<#6M>{mYCmq;AV)K6c#?) zfUh6G41f69D08F!`O66UuYU|+R{Dv#^Y_;oC`ZiSKP+{w#P*+`eef-w_3y70POZiv zuLUm{b=hj}&+Eb;ir*$NYHDhCnwy)a183jY)Ch(}Mh<^^{?Bib<#QWdz$SX==EkdD zXnM}l($dM*m5i5{$g*mWGKdJRXV2t=O)OaLar7kLP+GIXsdI5ZTS-$hASvm7a_;1$ 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zxDSl8IUy5Zod+g)2yU$J#>)JwC&BLS`@l*Oi2C))`|*Ybwn0cjf)rt#VA<2PDmgt0 z+_9TojCCo3qM|qN-P;SYm*}6?cLt~P)TvW+)8CCK5Dy^Qta^q?{KBRDJ*NEe1ygOC z7Viq5(9Aj8FNn)<1l-@WAenx=%S(=G;6y zxXiU|=v=c#Uqq>(_&46_(9JMaONL%W#qDnjIoS`8+!ovafJ87iJZ$H+i)R%%NGP4y u*|%95(d;K9X?Pr_O7efvw_e-Y_U6Ldu(DvzH~4%g5;Y|)#e4q$ +Assume p, then through several steps of reasoning, conclude q. + +*Eks:* + +If $n$ is odd, then $n²$ is odd + +Def: An integer n is #underline[even] if $exists k in ZZ, s.t., n=2k$ + +n is odd if $exists in ZZ #text("such that") n=2k+1$ + +Assume n is odd. then $exists k in ZZ #text("such that") n=2k+1$ + +So + +$ +n²=(2k+1)²=4k²+4k+1\ \ +=2(2k²+2k)+1 +$ +Since $k in ZZ$, then $2k²+2k in ZZ$, so $n²$ is odd. + + +== Proof by contraposition +Prove $p->q$ by proving $not p-> not q$ + +For an integer n, if n² is odd, then n is odd. + +Suppose n² is odd, then $exists k in ZZ #text("such that") n²=2k+1$ + +So $n=plus.minus sqrt(2k+1)$ + +We prove that if n is even, then n² is even. + +Suppose n is even, then $exists in ZZ #text("such that") n=2k$ + +So $n²=(2k²)=4k²=2(2k²)$ + +Since $2k²$ is an integer, n² is even. + +== Proof by contradiction +Want to prove _p_ by assuming it's not true (assume $not p$) and then reach a contradiction. + +By contradiction we are going to conclude something we know is false. + +Showing: $not p->p$ + +*Eks:* +$sqrt(2)$ is irrational + +A real number r is #underline[rational] if $exists a,b in ZZ$ #underline[with] $b eq.not 0 space #text("such that") space r=a/b$ + +Assume $sqrt(2)$ is rational. Then $exists a, b in ZZ, b eq.not 0 space #text("such that") space sqrt(2)=a/b$. You can assume that a and b have no common factors. + +Then: +$ +2=(a²)/(b²) \ +2b²=a² +$ +So a² is even, therefore a is even, then: +$ exists k in ZZ, a=2k $ +$ +2b²=(2k²)=4k²=> \ +b²=2k² +$ +So b² is even and therefore b is even. + +Now that we've shown that a and b have a common factor (they have a common factor, 2, because both are even). + +So now we have shown that $sqrt(2)$ can't be a rational number because there then is a contradiction. + += Sets +Def: A #underline[set] is an unordered collection of objects called #underline[elements]/#underline[members] of the set. We a set contains its elements. + +$a in A$ - a is an element of A + +$ a in.not A$ - a is NOT an element of A + +It's also called the naive set theory, which leads to some paradoxes (Russell's Paradox, exercise 50 in 2.1 in book). + +A way to describe a set, is to list all its elements. Eks: $A= {1,2,3,4}, B = {a,b,c,d}, C={"Apple",7,"Mike","Tivoli"}$ + +You can also do: ${x|"x has property P"}$ + +Ex: +$ O&={x|x" is an odd integer"} \ +&={x in ZZ|x "is odd"} +$ + +A set can have sets as elements. A set does not contain it's subelements if it contains sets. ${NN, ZZ, QQ, RR}$ does e.q. not contain the number 2. It contains sets that contain the number 2, but it itself does not. + +== Important sets +$NN={0,1,2,3,4,...}$ - Set of natural numbers (some argue 0 is not in the natural numbers, some argue it is). + +$ZZ={...,-2,-1,0,1,2,...}$ - Set of Integers. Can say $ZZ⁺$ or $ZZ_(>=0)$ for integers greater than 0. + +$QQ = {a/b|a,b in ZZ, b eq.not 0}$ - Set of rational numbers + +$RR$ - Real numbers (all numbers) + +$CC$ - Complex numbers + +== When two sets are equal +Def: Two sets A and B are equal if and only if they contain exactly the same elements. + +$A=B$ if and only if $forall x{x in A <-> x in B}$ + +Order and amount of appearances doesn't matter in sets, so $ {1,3,5}={5,3,1}={1,3,3,5,5,5} $ + +If you have a set of variables, and some of them are equal, their value still only appears once. + +== Empty set +This is the set with no elements. Notation $emptyset$ + +== Singleton set +Set with one element + +== Subset +Def: A is a #underline[subset] of B, and B is a #underline[superset] of A, if and only if, every element of A is an element of B. Denoted $A subset.eq B$ or $B supset.eq A$. + +You should say "A is a subset of B" + +To show $A subset.eq B$: show that $"if" x in A ", then" x in B$. Show that every element in A is in B + +To show $A subset.eq.not B$: find $"some" x in A "such that" x in B$. Find an element in A that is not in B. + +*Theorem:* For any set S, $emptyset subset.eq S$ and $S subset.eq S$ + +$forall x(x in nothing -> x in S)$ is true ($x in nothing$ will always be false, so that means the implication is always true). + +== Size of set (cardinality) +Def: Let S be the set. If there are exactly n distinct elements in S where $n in NN$, then we say that S is a finite set, with cardinality (size) n. $|S|=n$. + +$ +A&={x in ZZ⁺|"x is odd and" x <=10} \ +|A|&=5\ +|nothing|&=0 +$ +A set is infinite if it's not finite ($NN, RR, ZZ, QQ$ are for example all infinite). + +== Powerset +Def: Let S be a set, the #underline[powerset] of S, denoted $cal(P)(S)$, is the set of all subsets of S + +$cal(P)({0,1,2,3})={nothing,{0},{1},{2},{3},{0,1},{0,2}{1,2},{0,1,2,3}}$ + +$|S|=n , |cal(P)(S)|=2^n$ +$cal(P)(nothing)={nothing}$ + +== Cartesian products + +Def: The (ordered) n-tuple $(a_1,a_2,...,a_n)$ is the ordered collection that has $a_i$ as its i'th element. + +$(a_1,a_2,...,a_n)=(b_1,b_2,...,b_n) "if and only if" a_i=b_i forall i=1,2,...,n$ (n-tuples are equal if and only if their elements are the same and in the same order) + +2-tuples are called ordered pairs + +$(a,b) != (b,a)$ (unless $a=b$) + +*Def:* Let A and B be sets. The cartesian product of A and B, denoted $A times B$, is the set of all ordered pairs (a,b) where $a in A$ and $b in B$ + +$A times B={(a,b)|a in A and b in B}$ + +Ex: $A={1,2}, B={a,b,c}$ + +$ A times B={(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)} $ + +$A times B != B times A$ (unless $A=B$ or $A=nothing or B=nothing$) + +=== Cartesian product of many sets +Def: The Cartesian product of $A_1,A_2,...,A_n$, denoted by $A_1 times A_2 times ... times A_n$ is the set of n-tuples ($a_i,a_2,...,a_n$) such that $a_i in A_i "for all" i=1,2,...,n$ + + +*NOTE:* $A times (B times C) !=(A times B)times C !=A times B times C$ + +First one would be $((a,b),c)$, second one would be $(a,(b,c))$, third would be $(a,b,c)$. + +$A²=A times A,space A³=A times A times A$, etc. + +== Relations +A subset R of $A times B$ is called a relation from A to B. + +A relation from A to itself is called a relation on A. + +*Example* + +$A={0,1,2,3}$ + +$ +R&={(a,b)|a,b in A and a <=b} \ +&={(0,0),(0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)} +$ \ No newline at end of file diff --git a/Diskret Mat/Propositional and predicate logic/main.typ b/Diskret Mat/Propositional and predicate logic/main.typ new file mode 100644 index 0000000..ab5fbd0 --- /dev/null +++ b/Diskret Mat/Propositional and predicate logic/main.typ @@ -0,0 +1,118 @@ += Opgaver +#image("Opgaverne.png") +== Opgave 1.1-1 +#image("Opgave-1.1-1.png") +a) Proposition. And it is true + +b) Proposition. And it is false. + +c) Proposition. True + +d) Proposition. False + +e) Not a proposition because it has a variable. + +f) Not a proposition because it is a command. + +== Opgave 1.1-3 +#image("Opgave 1.1-3.png") +a) Linda is not younger (older or same age) than Sanjay + +b) Mei does not make more money than Isabella (makes less or same) + +c) Moshe is not taller than Monica (shorter or same age) + +d) Abby is not richer than Ricardo + + +== Opgave 1.1-11 +#image("Opgave-1.1-11.png") +a) Sharks have not been spotted near the shore. + +b) Swimming at the New Jersey shore is allowed and sharks have been spotted near the shore. + +c) Either swimming is not allowed at the New Jersey shore or sharks have been spotted near the shore. + +d) Swimming is allowed at the shore which implies sharks have not been spotted. + +e) Sharks have not been spotted at the shore which implies it is allowed to swim there. + +f) Swimming is not allowed at the shore which implies sharks have not been spotted. + +g) Swimming is allowed at the shore if and only if sharks have not been spotted. + +h) Swimming is allowed or sharks have not been spotted and swimming is not allowed. + +== Opgave 1.1-37 + +#image("Opgave 1.1-37.png") +#table(columns: 10)[p][q][$not p$][$not q$][$p->not q$][$not p <->q$][$(p->q)or(not p->q)$][$(p->q)and(not p-> q)$][$(p<->q)or(not p<->q)$][$(not p <-> not q)<->(p<->q)$][T][T][F][F][F][F][T][T][T][T][F][T][T][F][T][T][T][T][T][T][T][F][F][T][T][T][T][F][T][T][F][F][T][T][T][F][T][F][T][T] + + +== Opgave 1.1-53 +#image("Opgave 1.1-53.png") +a) Only the 99th statement would be true. For example the 3rd statement cant be true because if only 3 of the statements are false, that would mean the other 97 statements are true, but they cant be because they say other amounts of statements are false. But for the 99th, it would be true that all the other statements are false except for that one. The 100th statement cant be true because if all statements in the list are false, then the 100th would be true, which it cant be if all are false. + +b) Only the statements of $n<=50$ would could be true, because for the 51th statement "At least 51 of the statements are false" it is incorrect that _at least_ 51 of the statements are false (statements 1-50 are also true, so there's only 49 remaining false statements which is contradicting the statement). + +c) It would mean that statements 1-49 would be true as for the 49th statement, at least 49 of the statements _are true_ (real answer is 50 but 50 is at least 49). + +== Opgave 1.4-1 +#image("Opgave 1.4-1.png") +a) $P(0)=T$ + +b) $P(4)=T$ + +c) $P(6)=F$ + +== Opgave 1.4-31 +#image("Opgave 1.4-31.png") +a) $forall y Q(0,0,0) and forall y Q(0,1,0)$ + +b) $exists x Q(0,1,1) or exists x Q(1,1,1) or exists x Q(2,1,1)$ + +c) $exists z not Q(0,0,0) or exists z not Q(0,0,1)$ + +d) $exists x not Q(0,0,1) or exists x not Q(1,0,1) or exists x not Q(2,0,1)$ + +== Opgave 1.4-35 +#image("Opgave 1.4-35.png") +a) $exists x(x<=1)$ + +b) $exists x(x>2)$ + +c) $forall x(x<4)$ + +d) $forall x(x>=0)$ + +e) $exists x((x>=-1) and (x<=2))$ + +f) $forall x((x>=4) and (x<=7))$ + +== Opgave 1.4-37 + +#image("Opgave 1.4-37.png") +a) No counterexample, it would only be happen for numbers between 0 and 1 (0.1,0.2,...0.9) + +b) $exists x(x=0)$ + +c) $exists x(x>1or x<1)$ + +== Opgave 1.4-41 +#image("Opgave 1.4-41.png") +a) If there exists a printer that is out of service and is also busy, that leads to some work being lost. + +b) If all printers are busy, then a job is in the queue + +c) If there exists a print job that gets queued but also lost, then a printer is out of service + +d) If all printers are busy and all print jobs are queued, a print job has been lost + + +== Opgave 1.4-55 +#image("Opgave 1.4-55.png") +a) Truth + +b) False + +c) True \ No newline at end of file diff --git a/Diskret Mat/Sets and functions/Different 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zp%U)yKH{BNbdec;4ugx>K2}y9LJf{1pTO%kK|pI^yg^4$Vi3=lC0@w$@KCDqXjOP;FaVOzJ^$*>krBn~iTxqAk&ln>&Z4%k6OVn+e5cl|`< zu<-lqqzQv7jCQDKfYEH;_oX>\ +(dash(A)inter dash(B)) inter (dash(B) inter dash(C)) inter (dash(A) inter dash(C)) <=> \ +"We know because of the distributive laws, that in our case, the parenthases dont matter, so:"\ +dash(A) inter dash(B) inter dash(B) inter dash(C) inter dash(A) inter dash(C) <=> \ +"We can remove dupplicates:"\ +dash(A)inter dash(B) inter dash(C) +$ +#image("Opgave 2.2-47-prerequisits.png") +#image("Opgave 2.2-47.png") +Yes. If A contained an element that B did not, it would mean $A plus.circle B$ now contains elements not in $B plus.circle C$ because A has an extra element not in C, so it is in $A plus.circle C$, but since B does not contain it, it is not in $B plus.circle C$. A and B must also contain the same elements of C, since if they did not, no matter what $A plus.circle C = B plus.circle C$ would not be true. + +#image("opgave 2.3-9g.png") +$floor(1/2+ceil(3/2))<=>\ floor(1/2+2)=floor(2.5)=2$ + +#image("opgave 2.3-21.png") +a) $f(x)=2|x|+x$ - It will always be a positive even integer minus x. + +b) $f(x)=|x|$ - All positive integers are mapped to an integer, but they are all mapped to two. + +c) $underbracket(f\: ZZ arrow ZZ^+, f(x) = cases(2x "if" x > 0, -2x + 1 "if" x <= 0))$ + +d) $f(x)=x^2$ + +#image("opgave 2.3-27.png") +a) For a strictly decreasing function we know: $x>y=>f(x)>f(y) space forall x,y in A$, which means that two for two different x and y, f(x) and f(y) must be different. + +b) $f(x)=cases(x "if" x>=0,-1 "if" x<=0)$ + +#image("opgave 2.3-73.png") +#image("opgave 2.3-73a.png") +If one of $f_(A)(x) "or" f_(B)(x)$ is 0, then $f_(A inter B)(x)$ must be 0 because of the $inter$. $f_(A)(x) dot f_(B)(x)$ must also be 0 because then you are multiplying with 0, which will always be 0. \ No newline at end of file diff --git a/Diskret Mat/Sets and functions/Sets and functions.typ b/Diskret Mat/Sets and functions/Sets and functions.typ new file mode 100644 index 0000000..8700231 --- /dev/null +++ b/Diskret Mat/Sets and functions/Sets and functions.typ @@ -0,0 +1,169 @@ +=== Union of 2 sets +Def: Let A and B be sets. The Union of A and B, denoted by $A union B$, is the set that contains every element in either A or B or both. +$ A union B={x|x in A or x in B} $ + +The order of the operation doesn't matter + +#image("union-of-a-and-b.png", width: 50%) +=== Intersections +Def: Let A and B be sets. The intersection of A and B, denoted by $A inter B$, is the set containing those elements, that are in both A and B. +$ A inter B = {x|x in A and x in B} $ + +The order of the operation doesn't matter +#image("intersection.png",width: 50%) + +=== Disjoint +Def. A and B are disjoint if $A inter B =nothing$, aka they don't have any elements in common. + +=== Size of elements in two sets +If A and B are finite sets, then: $ |A union B| = |A| + |B| - |A inter B| $ +To get size of the elements in the sets, you need to add their size together, and then subtract all the elements they have in common. + +=== Difference +Def: Let A and B be sets. The Difference of A and B, denoted $A-B$, is the set containing those elements of A that are not in B. $ A-B={x|x in A and x in.not B} $ +Sometimes also denoted $A\\B$ +#image("difference-a-b.png",width: 50%) +If $A-B=nothing$. Either $A=B$ or $A subset B$ (A is a subset of B). + +=== Complement +Everything that is not in the set. But you need a universal set (in the pictures, the rectangle is the universal set). + +Def: Let U be the universal set. The complement of the set A, denoted by $macron(A)$, is the containing all the elements of U that are not in A. $macron(A)=U-A$ +#image("complement.png",width: 50%) +Example. +$ +U=NN,space space space A&={1,3,5,7,...}\ +dash(A)&={0,2,4,6,...}\ +U=ZZ, space space space dash(A)&={...,-2,-1,0,2,4,...} +$ +=== Set identities +#image("set-identities.png",width: 75%) +Prove: $dash(A union B) = dash(A) union dash(B)$ + +$ x in dash(A union B) <=>\ +x in.not A union B <=> \ +x in.not A and x in.not B <=>\ +x in dash(A) and x in dash(B) <=> \ +x in dash(A) union dash(B) $ +If x is in the complement of the union of A and B, then it must not be in the union of A and B. In other words, it is not in either A or B. This can be written as x being in the complement of A and the complement of B. This can be then be written as x is in the union of the complement of A and the complement of B. + +Prove: $dash(A union (B inter C))=(dash(C) union dash(B)) inter dash(A)$ +$ +dash(A union (B inter C)) <=>\ +dash(A) inter (dash(B union C)) <=>\ +dash(A) inter (dash(B) union dash(C)) +$ + +== Unions and intersections of an arbitrary number of sets +$union_(i=1)^(arrow.ccw.half)A_i={x|exists space i in {1,dots,b} text("such that") x in A_i}$ + +$inter_(i=1)^(arrow.ccw.half)A_i={x|forall space i in {1,dots,n} text("such that") x in A_i}$ + +$union_(i=1)^(infinity)A_i={x|exists space i in ZZ^+ text("such that") x in A_i}$ + +$inter_(i=1)^(infinity)A_i={x|forall space i in ZZ^+ text("such that") x in A_i}$ + +Example: +$A_i={1,2,3,dots,i}, A_1={1}, A_2={1,2}$ + +$union^(infinity)_(i=1)A_i=ZZ^+, inter_(i=1)^(infinity)A_i={1}$ + + += Functions +Def: Let A and B be nonempty sets. A function _f_ from A to B is an assignment of exactly one element of B to each element of A. We write $f(a)=b$ if b is the element of B that _f_ assigns to _a_. + +We write $f: A->B$ to denote that _f_ is a function from A to B. + +$(a,f(a))in A times B$ + +Def: Let $f:A->B$ + +*Domain* of _f_ is A + +*Codomain* of _f_ is B + +If $f(a)=b$, then b is the *image* of _a_ under _f_, and _a_ is a *preimage* of _b_. For every _a_ it matches a single _b_. + +*Range/Image* of _f_ the set of all images + +Example: $A={a,b,c,d}, space B={1,2,3,4}, space f:A->B$ + +$f(a)=1,space f(b)=2,space f(c)=3,space f(d)=4$ + +*Real-valued function* means that $text("codomain")=RR$ (real number) + +*Integer-valued function* means that $text("codomain")=ZZ$ (integers) + + +== Sum +Def: let $f_1 text("and") f_2$ be realvalued functions from A, then $f_1+f_2 text("and") f_1f_2$ are functions from A to $RR$ defined as: $ (f_1+f_2)(x)=f_1(x)+f_2(x)\ f_1f_2(x)=f_1(x)f_2(x) $ +*Example:* +$f_1,f_2:RR->RR space f_1(x)=x^2, space f_2(x)=x-x^2$ + +$ (f_1+f_2)(x)=x\ f_1f_2(x)=x^3-x^4 $ + + +*Def:* Let $f:A->B$. IF $S subset.eq A$, then the image of S under f, denoted $f(S)$ is the set ${b in B|exists a in S text("such that") f(a)=b}$ + +Example: $f:RR->RR space f(x)=x^2$ + +$S={-1,0,1}\ f(S)={0,1}$ + +== Injectivity and Surjectivity +#image("Different correspondence.png") +=== Injectivity +Def: A function $f:A->B$ is one-to-one/injective/an injection if $f(x)=f(y)=>x=y, forall x,y in A$. Two elements in the domain, cannot match to the same element in the codomain. + +$x!=y=>f(x)=>f(y)$. + +*Example* + +$f:ZZ->ZZ space f(x)=x^2$ + +This is not injective as for example both -1 and 1 give the same answer. + +$f:NN->ZZ space f(x)=x^2$ would be injective as you now only have positive x. + +~ + +*Def:* Let $f:A->B text("such that") A,B subset.eq RR$. Then _f_ is increasing if $x<=y=>f(x)<=f(y) space forall x,y in A$. + +It would be strictly increasing if $xf(x)B$ is surjective/onto/a surjection if $forall b in B, exists a in A "such that" f(a)=b$. For every element in the codomain, there exists an element in the domain that matches to it. + +Example: $f:ZZ->ZZ space f(x)=x-1$ + +Let $k in ZZ, "then" k+1 in ZZ "and" f(k+1)=k+1-1=k$. + +$f:NN->NN space f(x)=2x$ would not be surjective because the uneven numbers in the codomain would not have an element in the domain matched to them. $f:ZZ->ZZ space f(x)=2x$ would be surjective because for every real number, half that number would still be a real number, so every element in the codomain would have an element in the domain that matches it. + +=== Bijection +Def: A function is a bijection/one-to-one correspondence if it is both injective and surjective + +Suppose $f:A->B$ + +To show that f is _injective_: show if $f(x)=f(y), "then" x=y space("for arbitrary" x,y in A)$ + +To show that it's not _injective_: Find a particular $x,y in A "such that" x!=y "and" f(x)=f(y)$ + +To show that f is _surjective_: Consider arbitrary $b in B, "and show" exists a in A "such that" f(a)=b$ + +To show that it's not _surjective_: Find a particular $b in B "such that" exists.not a in A "with" f(a)=b$ + +== Inverse +Def: Let $f:A->B$ be a bijection. The inverse function of _f_, denoted $f^(-1)$, is the function from B to A that assigns to $b in B$ the unique element $a in A "such that" f(a)=b$. 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