根据图片生成视频---**Title:** 集合的包含关系 (Set Inclusion Relationship) **Definition:** 定义: 集合的包含关系用于描述两个集合之间的子集与超集关系。若集合A的所有元素都属于集合B, 则称A是B的子集, 或B是A的超集。 (Definition: Set inclusion relationship is used to describe the subset and superset relationship between two sets. If all elements of set A belong to set B, then A is called a subset of B, or B is called a superset of A.) **Formulas:** 公式: (Formulas:) · 子集: A ⊆ B (A是B的子集) (Subset: A ⊆ B (A is a subset of B)) · 真子集: A ⊂ B (A是B的真子集, 即B中至少有一个元素不在A中) (Proper subset: A ⊂ B (A is a proper subset of B, meaning B has at least one element that is not in A)) · 相等: A = B (当且仅当 A ⊆ B 且 B ⊆ A ) (Equality: A = B (if and only if A ⊆ B and B ⊆ A)) **Application Scenarios:** 应用场景: (Application Scenarios:) · 判断集合{1, 2}是否是{1, 2, 3}的子集 (是, 因为{1, 2}的所有元素都在{1, 2, 3}中) (Determine if set {1, 2} is a subset of {1, 2, 3} (Yes, because all elements of {1, 2} are in {1, 2, 3})) · 验证空集 ∅ 是任何集合的子集 (Verify that the empty set ∅ is a subset of any set) **Points to Note:** 注意事项: (Points to Note:) · 任何集合都是其自身的子集 ( A ⊆ A ), 但不是真子集 (Any set is a subset of itself ( A ⊆ A ), but not a proper subset) · 空集是任何集合的子集 (The empty set is a subset of any set) · 子集关系具有传递性: 若 A ⊆ B 且 B ⊆ C , 则 A ⊆ C (The subset relationship is transitive: If A ⊆ B and B ⊆ C, then A ⊆ C)