Set Definitions

Question 1

The 3 ways to write ∃!xP(x)

Answer 1

∃x (P(x) ∧ ∀y (P(y) → y = x) )

∃x ∀y (P(y) ↔ y = x)

∃x P(x) ∧ ∀y ∀z ( (P(y) ∧ P(z) ) → y = z)

Question 2

A⊆B

Answer 2

∀x (x ∈ A → x ∈ B)

Question 3

℘(A)

Answer 3

{x | x ⊆ A}

Question 4

A×B

Answer 4

{(a,b) | a ∈ A ∧ b ∈ B}

Question 5

Relation

Answer 5

R ⊆ A×B

Question 6

∩F

Answer 6

{x | ∀A (A ∈ F → x ∈ A)}

Question 7

∪F

Answer 7

{x | ∃A (A ∈ F ∧ x ∈ A)}

Question 8

Dom(R)

Answer 8

{a ∈ A | ∃b ∈ B ( (a,b) ∈ R)}

Question 9

Ran(R)

Answer 9

{b ∈ B | ∃a ∈ A ( (a,b) ∈ R)}

Question 10

R^-1

Answer 10

{(b,a) ∈ B×A | (a,b) ∈ R}

Question 11

SoR

Answer 11

{(a,c) ∈ A×C | ∃b ∈ B ( (a,b) ∈ R ∧ (b,c) ∈ S)}

Question 12

Reflexive

Answer 12

∀x ∈ A (xRx)

Question 13

Symmetric

Answer 13

∀x ∈ A ∀y ∈ A (xRy → yRx)

Question 14

Transitive

Answer 14

∀x ∈ A ∀y ∈ A ∀z ∈ A ( (xRy ∧ yRx) → xRz)

Question 15

Antisymmetric

Answer 15

∀x ∈ A ∀y ∈ A ( (xRy ∧ yRx) → x = y)

Question 16

Partial Order

Answer 16

Reflexive

Transitive

Antisymmetric

Question 17

Total Order

Answer 17

Partial Order

∀x ∈ A ∀y ∈ A (xRy ∨ yRx)

Question 18

R-smallest

Answer 18

∀x ∈ B (bRx)

Question 19

R-largest

Answer 19

∀x ∈ B (xRb)

Question 20

R-minimal

Answer 20

¬∃x ∈ B (xRb ∧ x ≠ b)

Question 21

R-maximal

Answer 21

¬∃x ∈ B (bRx ∧ x ≠ b)

Question 22

Equivalence Relation

Answer 22

Reflexive

Symmetric

Transitive

Question 23

Pairwise Disjoint

Answer 23

∀X ∈ F ∀Y ∈ F (X ≠ Y → X∩Y = ∅)

Question 24

Partition

Answer 24

∪F = A

Pairwise Disjoint (∀X ∈ F ∀Y ∈ F (X ≠ Y → X∩Y = ∅))

∀X ∈ F (x ≠ ∅)

Question 25

Equivalence Class

Answer 25

[x]_R = {y ∈ A | yRx}

Question 26

A/R

Answer 26

{[x]_R | x ∈ A}

Question 27

x is congruent to y (mod m)

(modulo)

Answer 27

∃k ∈ Z (x - y = km)

Question 28

f: A → B

Answer 28

∀a ∈ A ∃!b ∈ B ( (a,b) ∈ f)

Question 29

f: A → B is one-to-one.

Answer 29

¬∃a₁ ∈ A ∃a₂ ∈ A ( f(a₁) = f(a₂) ∧ a₁ ≠ a₂)