Subsets, Proper Subsets and Supersets | Don't Memorise

Summary

This video explains the concepts of subsets, proper subsets, and supersets using simple examples with sets 'P', 'Q', 'C', and 'D'. It demonstrates how set 'P' is a subset of 'Q', and introduces the concept of proper subsets, where 'P' is a proper subset of 'Q' because it contains fewer elements. The video also clarifies that every set is a subset of itself, but a proper subset must have at least one element missing from the larger set. Finally, it discusses the notation for proper subsets and supersets, enhancing understanding of these foundational set theory concepts.

Takeaways

• 😀 Set 'P' is a subset of set 'Q' when all elements of 'P' are also elements of 'Q'.
• 😀 Every set is a subset of itself, meaning 'P' is a subset of 'P', and 'Q' is a subset of 'Q'.
• 😀 The subset symbol (⊆) is used to represent that one set is a subset of another set.
• 😀 A proper subset (⊂) means that all elements of 'P' are in 'Q', but 'Q' contains at least one element not in 'P'.
• 😀 Set 'P' can be a proper subset of set 'Q' if 'Q' has elements that are not in 'P'.
• 😀 'C' is not a proper subset of 'D' because there is no element in 'D' not found in 'C'.
• 😀 A proper subset is also referred to as a 'strict subset'.
• 😀 The super set symbol (⊇) indicates that one set contains all elements of another set.
• 😀 Set 'C' and set 'D' are equal because they contain exactly the same elements.
• 😀 Subsets and proper subsets are foundational concepts in set theory, essential for understanding the relationships between sets.

Q & A

What is a subset?

A subset is a set where every element in one set is also present in another set. For example, if set 'P' = {1, 2, 3} and set 'Q' = {1, 2, 3, 4, 5}, then 'P' is a subset of 'Q' because all elements in 'P' (1, 2, 3) are in 'Q'.

What does the symbol '⊆' represent?

'⊆' represents a subset. For example, 'P ⊆ Q' means that set 'P' is a subset of set 'Q'.

What is the difference between a subset and a proper subset?

A subset can be equal to the set it is part of, while a proper subset must contain all the elements of the set but must have at least one element missing from the superset. For example, 'P' = {1, 2, 3} is a proper subset of 'Q' = {1, 2, 3, 4, 5}, but 'Q' is not a proper subset of 'P'.

How can we represent a proper subset using a symbol?

A proper subset is represented by the symbol '⊂'. For example, 'P ⊂ Q' means that 'P' is a proper subset of 'Q'.

Is every set a subset of itself?

Yes, every set is a subset of itself. This is because all elements in a set are obviously present in that same set.

Can a set be both a subset and a proper subset of another set?

A set cannot be both a subset and a proper subset of the same set simultaneously. A proper subset requires that there be at least one element in the superset that is not in the subset.

When is a set considered a proper subset?

A set 'P' is a proper subset of 'Q' if all elements in 'P' are in 'Q' and 'Q' contains at least one element not in 'P'. For example, 'P' = {1, 2, 3} is a proper subset of 'Q' = {1, 2, 3, 4, 5}, since 'Q' has the extra elements 4 and 5.

What is a superset?

A superset is the opposite of a subset. If 'P' is a subset of 'Q', then 'Q' is a superset of 'P'. For example, if 'P' = {1, 2, 3} and 'Q' = {1, 2, 3, 4, 5}, then 'Q' is a superset of 'P'. The symbol for a superset is '⊇'.

What does the symbol '⊇' represent?

'⊇' represents a superset. For example, 'Q ⊇ P' means that 'Q' is a superset of 'P'.

What is the key difference between a subset and a proper subset?

The key difference is that a proper subset requires at least one element in the superset that is not in the subset, while a subset can be the same as the superset. For example, 'P' = {1, 2, 3} is a subset of 'Q' = {1, 2, 3, 4, 5}, but 'P' is also a proper subset because 'Q' has extra elements (4, 5).