Subset Calculator

What is a Subset?

Imagine you have a big box of toys (let's call it Set B) and a smaller box of toys (Set A). If all the toys in the smaller box are also in the big box, we say that Set A is a subset of Set B. It's like the smaller box fits completely inside the bigger box!

How to Calculate if A is a Subset of B

To check if Set A is a subset of Set B, we need to make sure that every single toy (or element) in Set A is also in Set B. It's like checking off a list to make sure nothing is missing!

Formula

We can write the subset relationship like this:

\[ A \subseteq B \iff \forall x (x \in A \implies x \in B) \]

Where:

• \(A \subseteq B\) means "A is a subset of B"
• \(\iff\) means "if and only if"
• \(\forall\) means "for all"
• \(x\) represents an element
• \(\in\) means "is an element of"
• \(\implies\) means "implies" or "leads to"

Calculation Steps

1. Write down all the elements in Set A
2. Write down all the elements in Set B
3. Look at each element in Set A
4. Check if that element is also in Set B
5. If you find an element from A that's not in B, stop! A is not a subset of B
6. If all elements from A are in B, then A is a subset of B

Example and Visual Representation

Let's say we have two sets of animals:

Set A = {cat, dog}

Set B = {cat, dog, fish, bird}

To check if A is a subset of B, we look at each animal in Set A:

• Is "cat" in Set B? Yes!
• Is "dog" in Set B? Yes!

Since all animals in Set A are also in Set B, we can say that A is a subset of B!

We can show this with a picture:

In this picture, the red circle (Set A) is completely inside the blue circle (Set B), showing that A is a subset of B!