Geometry L5 Chapter 1 & 2 Review #1

Set Operations and Notation

• Given sets A, B, and C defined with specific conditions:
  • A = {x | x ≥ 2}
  • B = {x | -4 ≤ x ≤ 5}
  • C = {x | x < 1}

Operations on Sets

• Intersection (A ∩ B): Represents the set of elements common to both A and B.

  • Result: 2 ≤ x ≤ 5

• Union (A ∪ B): Represents the set of all elements that are in either A or B.

  • Result: -4 ≤ x

• Intersection of C and B (C ∩ B): Elements common to both C and B.

  • Result: x < 1

• Complement of A (denoted as A')(A ∩ B): Elements not in A and that are also in B.

  • Result: {x | (x < 2) or (x > 5)}

• Complement of B (B'): Elements not in B intersect with A.

  • Result: x < -4

• Intersection of C and B Complement (C ∩ B'):

  • Result: {x | (-4 ≤ x ≤ -1) or (1 ≤ x ≤ 5)}

Set Relationships

• Given conditions A - C - B and D - A - C.

Determining New Sets

• Intersection of AB and BC: Represents the elements common to both AB and BC.

  • Result: AB

• Intersection of AD and CB: Represents common elements between AD and CB.

  • Result: ∅ (empty set)

• Union of DA and AC: Represents all elements from DA and AC.

  • Result: DC

• Union of AD and AB: Represents all elements from AD and AB.

  • Result: BD

Geometric Representation

• For point P at coordinate -4, the expression for all points C within 4 units:
  • Expression: -4 - x ≤ 4 (geometric set described as a segment)

Graphical Representation of Sets

• Set {x | x + 1 ≤ 3}: Represents a segment on the number line.
• Set {x | x + 1 ≤ 0}: Represents a point on the number line.
• Set {x | (2x + 3) ≥ (x - 1)}: Represents a ray starting from a certain point onward.
• Set {x | (x + 1) ≥ -2}: Represents a ray.
• Set {x | x + 3 ≤ -2}: Represents an empty set (∅).

These core points summarize operations involving sets, their relationships, and representations geometrically on a number line.