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### 105 Cards in this Set

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 Set Identities A⋃∅ A⋂U Identity Laws A A Set Identities A⋃U A⋂∅ Domination Laws U ∅ Set Identities A⋃A A⋂A Idempotent Laws A A Set Identities A⋃B A⋂B Commutative Laws B⋃A B⋂A Set Identities A⋃(B⋃C) A⋂(B⋂C) Associative Laws (A⋃B)⋃C (A⋂B)⋂C Set Identities A⋂(B⋃C) A⋃(B⋂C) Distributive Laws (A⋂B)⋃(A⋂C) (A⋃B)⋂(A⋃C) Set Identities (A⋂B)⋃(A⋂C) (A⋃B)⋂(A⋃C) Reverse Distributive Laws A⋂(B⋃C) A⋃(B⋂C) Set Identities ~(A⋃B) ~(A⋂B) Those should be lines over the whole top De Morgan's Laws ~A⋂~B ~A⋃~B Those should be lines over individual letters Set Identities ~A⋂~B ~A⋃~B Those should be lines over individual letters Reverse De Morgan's Laws ~(A⋃B) ~(A⋂B) Those should be lines over the whole top Set Identities A⋃(A⋂B) A⋂(A⋃B) Absorption Laws A A Set Identities A⋃~A A⋂~A Should be lines on top Complement Laws U ∅ Truth sets {x|P(x)} explain the parts The elements of the set for which the values of x make the second part true - the elementhood test y∈{x|P(x)} P(y) y∉{x|P(x)} ~P(y) A is a truthset for the proposition P(x) What is A in set form What is meant by y∈A y∉A P(y) ~P(y) Redefine {x∈U|P(x)} {x|x∈U^P(x)} Redefine {x|x∈U^P(x)} {x∈U|P(x)} Redefine y∈{x∈U|P(x)} y∈U^P(y) Redefine: What is it? A⋂B It is a set {x|x∈A^x∈B} If A and B have truth tests, then it could mean P(x)^Q(x) What is it? x ∈ A⋂B It is a statement x∈A^x∈B and if they had propositions P(x)^Q(x) Redefine: What is it? A⋃B It is a set {x|x∈A v x∈B} If A and B have truth tests, then it could mean P(x) v Q(x) What is it? x ∈ A⋃B It is a statement x∈A v x∈B and if they had propositions P(x) v Q(x) Redefine: What is it? A/B It is a set {x|x∈A ^ x∉B} If A and B have truth tests, then it could mean P(x) ^ ~Q(x) What is it? x ∈ A/B It is a statement x∈A ^ x∉B and if they had propositions P(x) ^ ~Q(x) Redefine: {x|x∈A^x∈B} A⋂B Redefine: x∈A^x∈B x ∈ A⋂B Redefine: {x|x∈A v x∈B} A⋃B x∈A v x∈B x ∈ A⋃B Redefine: {x|x∈A ^ x∉B} A/B Redefine: x∈A ^ x∉B x ∈ A/B What is it? Name it Redefine: A∆B (2 definitions) It's a set Symmetric Difference (A\B)⋃(B\A) (A⋃B)\(A⋂B) Redefine: what is it? A⊆B (2 definitions) It is a statement: ∀x(x∈A → x∈B) ∀x∈A(x∈B) Redefine: What is it? A=B (2 definitions) It is a Statement ∀x(x∈A ↔ x∈B) A⊆B ^ B⊆A Redefine: (A\B)⋃(B\A) (A⋃B)\(A⋂B) A∆B Redefine: ∀x(x∈A → x∈B) ∀x∈A(x∈B) A⊆B Redefine: ∀x(x∈A ↔ x∈B) A⊆B ^ B⊆A A=B What is it? What is it called? Redefine A⋂B=∅ A statement Called disjoint Allx, x is part of the first part iff x is part of the second part? or This first part is a subset of the second, and the second is a subset of the first? What is it? Re-express: {pi|i∈I} An indexed set, the set contains all numbers pi with i being the element of some set. {x|∃i∈I(x=pi)} What is it? Re-express: {x|∃i∈I(x=pi)} An indexed set, the set contains all numbers pi with i being the element of some set. {pi|i∈I} An indexed set, the set contains all numbers pi with i being the element of some set. Give two ways {x|∃i∈I(x=pi)} {pi|i∈I} Re-express the family of sets F in bracket notation: using the classes of students Cs give two ways Each Cs is a set itself {{1 2},{3,4},{5,6}} C1 = {1,2} F = {Cs|s∈S} {X|∃s∈S(X=Cs)} What is it? Re-express: F = {Cs|s∈S} Indexed Family Each Cs is a set itself {{1 2},{3,4},{5,6}} C1 = {1,2} {X|∃s∈S(X=Cs)} What is it? Re-express: F = {X|∃s∈S(X=Cs)} Indexed Family Each Cs is a set itself {{1 2},{3,4},{5,6}} C1 = {1,2} F = {Cs|s∈S} What is it? Re-express P(A) The power set: {x|x⊆A} x is a set What is it? Re-express {x|x⊆A} x is a set The power set: P(A) The family set is a subset of what? Give reasoning If all courses were set C then each Cs is a subset of C For each student Cs∈P(C) Every element then of the family is an element of P(c) so F ⊆ P(C) Equivalence P(A⋂B) P(A)⋂P(B) Equivalence P(A)⋂P(B) P(A⋂B) Explain and define: ⋂F F = {{1,2},{2,3},{3,4}} this is the intersection of the family {1,2}⋂{2,3}⋂{3,4} {x|∀A∈F(x∈A)} {x|∀A(A∈F → x∈A)} What is this?/ {x|∀A∈F(x∈A)} F = {{1,2},{2,3},{3,4}} this is the intersection of the family {1,2}⋂{2,3}⋂{3,4} {x|∀A∈F(x∈A)} {x|∀A(A∈F → x∈A)} What is this? {x|∀A(A∈F → x∈A)} F = {{1,2},{2,3},{3,4}} this is the intersection of the family {1,2}⋂{2,3}⋂{3,4} {x|∀A∈F(x∈A)} {x|∀A(A∈F → x∈A)} What is this? {1,2}⋂{2,3}⋂{3,4} F = {{1,2},{2,3},{3,4}} this is the intersection of the family {1,2}⋂{2,3}⋂{3,4} {x|∀A∈F(x∈A)} {x|∀A(A∈F → x∈A)} Explain and define: ⋃F F = {{1,2},{2,3},{3,4}} this is the union of the family {1,2}⋃{2,3}⋃{3,4} {x|∃A∈F(x∈A)} {x|∃A(A∈F ^ x∈A)} What is this?/ {x|∃A∈F(x∈A)} F = {{1,2},{2,3},{3,4}} this is the union of the family {1,2}⋃{2,3}⋃{3,4} {x|∃A∈F(x∈A)} {x|∃A(A∈F ^ x∈A)} What is this? {x|∃A(A∈F ^ x∈A)} F = {{1,2},{2,3},{3,4}} this is the union of the family {1,2}⋃{2,3}⋃{3,4} {x|∃A∈F(x∈A)} {x|∃A(A∈F ^ x∈A)} What is this? {1,2}⋃{2,3}⋃{3,4} F = {{1,2},{2,3},{3,4}} this is the union of the family {1,2}⋃{2,3}⋃{3,4} {x|∃A∈F(x∈A)} {x|∃A(A∈F ^ x∈A)} Define ⋂F=∅ undefined What is it? Redefine ⋂(i∈I) Ai Where F puts Ai as sets in a set, this is more like the union of all the sets, it is the same thing as ⋂F {x|∀i∈I(x∈Ai)} {x|∀i(i∈I → x∈Ai)} It asks that all the sets have the same element What is it? {x|∀i∈I(x∈Ai)} ⋂(i∈I) Ai Where F puts Ai as sets in a set, this is more like the union of all the sets, it is the same thing as ⋂F {x|∀i∈I(x∈Ai)} {x|∀i(i∈I → x∈Ai)} It asks that all the sets have the same element What is it? Redefine ⋃(i∈I) Ai Where F puts Ai as sets in a set, this is more like the union of all the sets, it is the same thing as ⋃F {x|∃i∈I(x∈Ai)} {x|∃i(i∈I ^ x∈Ai)} That at least one of the sets of Ai have the element What is it? Redefine {x|∃i∈I(x∈Ai)} Where F puts Ai as sets in a set, this is more like the union of all the ⋃(i∈I) Ai How is this different from a indexed set or indexed family? Because it's an element, not equal sets, it is the same thing as ⋃F {x|∃i∈I(x∈Ai)} {x|∃i(i∈I ^ x∈Ai)} That at least one of the sets of Ai have the element What is it? Define AxB Cartesian Product {(a,b)| a∈A ^ b∈B} results in all ordered pairs What is it? Define {(a,b)| a∈A ^ b∈B} Cartesian Product AxB results in all ordered pairs Equivalence: Ax(B⋂C) (AxB)⋂(AxC) Equivalence: (AxB)⋂(AxC) Ax(B⋂C) Equivalence: Ax(B⋃C) (AxB)⋃(AxC) Equivalence: (AxB)⋃(AxC) Ax(B⋃C) Equivalence: (AxB)⋂(CxD) (A⋂C)x(B⋂D) Equivalence: (A⋂C)x(B⋂D) (AxB)⋂(CxD) Subset: (AxB)⋃(CxD) ⊆ (A⋃C)x(B⋃D) Equivalence: Ax∅ ∅xA ∅ AxB = BxA if and only if A =∅ or B = ∅ or A=B Truth set of P(x,y) two versions {(a,b)∈AxB | P(x,y)} {(a,b) | (a,b)∈AxB ^ P(x,y)} {(a,b)∈AxB | P(x,y)} Truth set of P(x,y) {(a,b) | (a,b)∈AxB ^ P(x,y)} Truth set of P(x,y) Define a relation from A to B R ⊆ AxB does not need to imply a truth set for R, it can be any subset What is this? R ⊆ AxB A relation Domain of R for AxB {a∈A | ∃b∈B((a,b)∈R} {a∈A | ∃b∈B((a,b)∈R} Domain of R Range of R {b∈B | ∃a∈A((a,b)∈R} {b∈B | ∃a∈A((a,b)∈R} Range of R Inverse of R R-1 {(b,a)∈BxA | (a,b)∈R} {(b,a)∈BxA | (a,b)∈R} Inverse of R R-1 Composition of two relations R⊆AxB S⊆BxC SoR {(a,c)∈AxC | ∃b∈B((a,b)∈R ^ (b,c)∈S} {(a,c)∈AxC | ∃b∈B((a,b)∈R ^ (b,c)∈S} Composition of two relations R⊆AxB S⊆BxC SoR (r,s)∈L-1 if and only if (s,r)∈L (s,p)∈ToE if and only if ∃c( (s,c)∈E ^ (c,p) ∈ T (x,y) ∈ R xRy xRy (x,y) ∈ R Relation equivalence: (R-1)-1 R Relation equivalence: Relation equivalence:Dom(R-1) Ran(R) Relation equivalence: Ran(R) Dom(R-1) Relation equivalence: Ran(R-1) Dom(R) Relation equivalence: Dom(R) Ran(R-1) Relation equivalence: To(SoR) (ToS)oR Relation equivalence: (ToS)oR To(SoR) (SoR)-1 R-1 o S-1 Equivalence Relation: R-1 o S-1 (SoR)-1 the relation R is reflexive on A (or just reflexive if A is clear from context) if (2 versions) ∀x∈A(xRx) or ∀x∈A((x,x)∈Rx) the relation R is symmetric if ∀x∈A∀y∈A(xRy → yRx) or ∀x∈A∀y∈A((x,y)∈R → (y,x)∈R) The relation R is transitive if ∀x∈A∀y∈A∀z∈A ((xRy ^ yRz) → xRz) also can be done without shorthand R is reflexive iff i(A) ⊆ R, where as before i(A) is the identity relation on A R is symmetric iff R=R-1 R is transitive iff RoR ⊆ R ∃!xP(x) 4 definitions ∃x(P(x)^~∃y(P(y) ^ y≠x)) ∃x(P(x)^∀y(P(y)→y=x)) ∃xP(x) ^ ∀y∀z[(P(y)^P(z))→y=z] Existence Uniqueness ∃x∀y(P(y)↔y=x)