12 B ⇥ Q = { ( b,q ) 2 R | b 2 B,q 2 Q } • A ⇥ P = { ( a,p ) 2 R | a 2 A,p 2 P }• R : • Ordermattersfor A ⇥ A ,if x,y 2 A and x 6 = y ,then ( x,y ) 6 =( y,x ) .Consider • A ⇥ B 6 = B ⇥ A . • Important:Ordermatters. A ⇥ B ⇥ ... ⇥ Z = { ( a,b,...,z ) | a 2 A,b 2 B,...,z 2 Z } . Givensets A,B,...,Z ,theircartesianproductisthesetoforderedtuples: 7.4CartesianProducts • Then 9 y 2 A s.t. y 2 A and y/ 2 B . A * B : • i.e.Proveallelementsin A arein B andviceversa.Toprove Toprovesets A = B thenshow ( A ⇢ B ) ^ ( B ⇢ A ) . 7.3ProofTechniquesforStatementsaboutSets • or: ( A \ B ) = A [ B b. A \ B =( A [ B ) • or: ( A [ B ) = A \ B a. A [ B =( A \ B ) Proposition18: DeMorgan’sLaws ( A ⇥ B ) \ ( D ⇥ E )=( A \ D ) ⇥ ( B \ E ) . Proposition17: Considersets A,B , D and E .Then:

Proposition 17: Consider setsA, B,DandE. Then:

(A⇥B)\(D⇥E) = (A\D)⇥(B\E).

Proposition 18: De Morgan’s Laws a. A^C[B^C= (A\B)^C

• or: (A^C[B^C)^C=A\B b. A^C\B^C= (A[B)^C

• or: (A^C\B^C)^C=A[B

7.3 Proof Techniques for Statements about Sets

To prove setsA=B then show(A⇢B)^(B⇢A).

• i.e. Prove all elements inAare inBand vice versa.

To proveA*B:

• Then9y2As.t. y2Aandy /2B.

7.4 Cartesian Products

Given setsA, B, ..., Z, their cartesian product is the set of ordered tuples:

A⇥B⇥...⇥Z={(a, b, ..., z)|a2A, b2B, ..., z2Z}.

• Important: Order matters.

• A⇥B6=B⇥A.

• Order matters forA⇥A, ifx, y2Aandx6=y, then(x, y)6= (y, x).

ConsiderR²:

• A⇥P ={(a, p)2R²|a2A, p2P}

• B⇥Q={(b, q)2R²|b2B, q2Q}

12

Consider the cartesian product of a setS with itself repeatedktimes,S^k: S^k= S⇥S⇥...⇥S

| {z }

k times ={(s1, s2, ..., sk)|si2S, 8i2{1,2, ..., k}}

• The elements= (s1, s2, ..., sk)is a vector.

• Eachsi2S fori2{1, ..., k}is a co-ordinate of vectors.

• The diagonal of setS^kis the following subset ofS^k;{(x, x, ..., x)2S^k|x2 S}.

– E.g. If A = {1,2,3}, then the diagonal of A³ = A⇥A⇥A = {(a, a, a)2A³|a2A}={(1,1,1),(2,2,2),(3,3,3)}.

7.5 Euclidean Spaces

An example of a cartesian product is the Euclidean spaceRⁿ=R⇥R⇥...⇥R for somen2N^⇤.

• ConsiderRⁿ, each elementx2Rⁿis a point or a vector of spaceRⁿ.

• Every vectorx2Rⁿis written as: x= (x1, x2, ..., xn)2Rⁿ. Vector operations:

• (Summation) Let vectorsx, y2Rⁿ, thenx+y= (x₁+y₁, x₂+y₂, ..., x_n+ yn).

• ( x) Let 2Randx2Rⁿ, then x= ( x1, x2, ..., xn).

Canonical vectors: Considerx2Rⁿ, the vectorxcan be decomposed as:

x=Pn

i=1x_ie_i=x₁e₁+...+x_ne_n

wheree_i2Rⁿbut are 0’s all but thei^thterm equals 1, the canonical vector.

• Canonical decomposition: The decompositionx=x1e1+...+xnen. Definition: x >> y () xi> yi8i2{1, ..., n}

Definition: x > y () (xi yi)^(9xk> yk)

• ‘at least one greater than’

Definition: x y () (x_i y_i)

• It is possible thatxi=yi8i={1,2, ..., n}.

13

7.6 Power Sets

Let A be a set. Then the power set, denoted 2^A is defined as the set of all subsets ofA.

• If setAhask elements, then the power set2^Ahas2^k elements.

8 Partition of a Set

Definition: Partition of a set

A (finite) partition of a non-empty setS is a finite collection of non-empty subsetsB1, B2, ..., Bk⇢S, for somek2N^⇤, such that:

Bi\Bj=;, ifi6=j and

S=Sk i=1Bi.

• B={B1, B2, ..., Bk}is a partition ofS.

• B is a set whose elements are subsets decomposing setS.

Definition: Refinement

Consider two partitionsBandCof a non-empty setS. If for every element Ci2C, there are elementsBj1, ..., Bjs2Bsuch that:

Ci=Ss r=1Bjr

thenBrefinesCor equivalentlyBis a refinement ofC.

• IfB refinesCthenCis a coarsening ofB.

• Any partitionB ofSrefines itself.

• IfBrefinesCandB refinesD, thenB is a common refinement ofCand D.

• IfBrefinesCand they are different partitions, thenB is a proper refine- ment ofC.

Example 29: SupposeS ={1,2,3,4,5}. Consider the following 3 partitions of S: R=P={{1,2},{3,4,5}}andQ={{1},{2},{3,4,5}}.

• Qis a proper refinement ofRandP.

• Qis a common refinement ofRandP.

• RandP are proper coarsening ofQ.

• P is a refinement ofR, but not a proper refinement.

• P is a coarsening ofR, but not a proper coarsening.

• P is a common coarsening ofRandQ.

14