Generalized Intersections of Sets ∩S 些{x VTES.xeT} ∩S(i) def ∩{S()|i∈} iel 50些∩50 i三m ie m,n 8/40
Generalized Intersections of Sets T S def = {x | ∀T ∈ S. x ∈ T} T i∈I S(i) def = T {S(i) | i ∈ I } Tn i=m S(i) def = T i∈[m,n] S(i) 8 / 40
Generalized Unions and Intersections of Empty Sets From USgf{xI3T∈S.xeT} ∩Sg{x|TeS.xeT} we know U0=0 ∩0 meaningless 0is meaningless,since it denotes the paradoxical "set of everything". 9/40
Generalized Unions and Intersections of Empty Sets From S S def = {x | ∃T ∈ S. x ∈ T} T S def = {x | ∀T ∈ S. x ∈ T} we know S ∅ = ∅ T ∅ meaningless T ∅ is meaningless, since it denotes the paradoxical ”set of everything”. 9 / 40
Outline Sets Relations Functions Products Sums 10/40
Outline Sets Relations Functions Products Sums 10 / 40
Relations We need to first define the Cartesian product of two sets A and B: A×B={(x,y)Ix∈A and y∈B} Here (x,y)is called a pair. Projections over pairs: πo(x,y）=xand1(x,y)=y. Then.p is a relation from A to B if pC Ax B,or pEP(Ax B) 11/40
Relations We need to first define the Cartesian product of two sets A and B: A × B = {(x, y) | x ∈ A and y ∈ B} Here (x, y) is called a pair. Projections over pairs: π0(x, y) = x and π1(x, y) = y. Then, ρ is a relation from A to B if ρ ⊆ A × B, or ρ ∈ P(A × B). 11 / 40
Relations p is a relation from A to B if pAx B,orp∈P(A×B). p is a relation on S if pSx S. We say p relates x and y if(x,y)Ep.Sometimes we write it as xpy. p is an identity relation if∀x,y）∈p.x=y. 12/40
Relations ρ is a relation from A to B if ρ ⊆ A × B, or ρ ∈ P(A × B). ρ is a relation on S if ρ ⊆ S × S. We say ρ relates x and y if (x, y) ∈ ρ. Sometimes we write it as x ρ y. ρ is an identity relation if ∀(x, y) ∈ ρ. x = y. 12 / 40