1.2.2
Subsets
Subsets
Subsets
As we have seen previously, the universal set contains all elements, and can also contain sets. We say that these sets are subsets of the universal set.
![Illustrative background for Subsets](https://image-v2.cdn.app.senecalearning.com/2019-10/9c4ec48c-e3b5-4ba6-86a5-9a13e808b10f/subset-min,h_400,q_80,w_640.png)
![Illustrative background for Subsets ?? "content](https://image-v2.cdn.app.senecalearning.com/2019-10/9c4ec48c-e3b5-4ba6-86a5-9a13e808b10f/subset-min,h_400,q_80,w_640.png)
Subsets
Subsets
- Subsets are sets contained within other sets.
- The set A is a subset of set B if all of the elements in A are also elements in B.
- For example the set A = {2, 3, 4} has the following subsets:
- {2, 3}, {3, 4}, {2, 4}, {2}, {3}, {4}, {2,3,4}
- Every set is its own subset.
![Illustrative background for Proper subsets](https://image-v2.cdn.app.senecalearning.com/2020-04/7e3b8a8f-f078-45f3-b948-0b70b65dee3c/Screenshot%202020-04-23%20at%2016.51.03-min,h_400,q_80,w_640.png)
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Proper subsets
Proper subsets
- Proper subsets of the set A are all the subsets that do not equal A.
- For example, the set A = {2, 3, 4} has the following proper subsets:
- {2, 3}, {3, 4}, {2, 4}, {2}, {3}, {4}
- {2, 3, 4} is not a proper subset of A.
![Illustrative background for Notation](https://image-v2.cdn.app.senecalearning.com/2019-10/fbc5fe87-f6dd-4b1b-b08b-4d6aa66fd00b/subset%20notation-min,h_400,q_80,w_640.png)
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Notation
Notation
- If A is a subset of B, we write:
- A ⊆ B
- If A is a propersubset of B, we write:
- A ⊂ B
- If C is not a subset of B, we write:
- C ⊄ B
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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