Sets (Junior Cert Mathematics): Revision Notes

Sets

What is a Set?

Imagine a set as a group or collection of things that have something in common. For example, think about a box of coloured pencils. If you pick out only the red pencils, you now have a set of red pencils. In maths, a set is simply a collection of items, called elements.

Here, the elements of the set are the vowels, and they are listed inside curly brackets $\{\}$.

Universal Set (U)

The Universal Set, usually written as $U$, is like the big box that contains everything we might be interested in. If we're talking about letters, the Universal Set could be all the letters in the alphabet.

Creating Specific Sets

From the Universal Set, we can create smaller sets by picking out certain elements. These smaller sets might represent different groups or categories.

Important Set Symbols

As we start to work with sets, we need to know some symbols that help us understand the relationships between them. Let's introduce these one at a time.

1. Intersection ($∩$)

• Meaning: The intersection of two sets is where the sets overlap. It includes only the elements that are in both sets.

Imagine: If set $A$ is the group of people who like pizza and set $B$ is the group of people who like burgers, the intersection $A \cap B$ is the group of people who like both pizza and burgers.

2. Union ($∪$)

• Meaning: The union of two sets is a new set that contains all the elements from both sets. It's like putting two sets together and not repeating any elements.

Imagine: If $A$ is people who play soccer and $B$ is people who play basketball, the union $A \cup B$ is the group of people who play either soccer, basketball, or both.

3. Difference ( \ )

• Meaning: The difference between two sets $A$ and $B$ (written as $A \setminus B$) is the set of elements that are in $A$ but not in $B$.

Imagine: If set $A$ is people who like chocolate and set $B$ is people who like vanilla, $A \setminus B$ is the group of people who like chocolate but not vanilla.

Further Set Symbols

Let's add some more symbols that you'll come across as you learn about sets.

4. Subset ( $⊆$ )

• Meaning: $A$ subset is a set where every element in it is also in another set. If $C$ is a subset of $A$, every element in $C$ is also in $A$.

Imagine: If everyone who likes apples also likes fruit in general, then the set of apple-lovers is a subset of the set of fruit-lovers.

5. Complement ( ' )

• Meaning: The complement of a set $A$ (written as $A'$) includes everything that is in the Universal Set $U$ but not in $A$.

Imagine: If $A$ is the set of students who like maths, then$A'$ is the set of students who don't like maths.

6. Null Set ( $\emptyset$ or $\{\}$ )

• Meaning: The null set is a set with no elements in it. It's empty!

7. Cardinal Number ( $\#$ )

• Meaning: The cardinal number of a set is the number of elements in the set.

Imagine: If set $A$ is the group of friends who came to your birthday party, and there were $5$ friends, then the cardinal number of $A$ is $5$.

Properties of Sets

Sets have some special rules that tell us how they behave when we combine them.

Commutative Property

• Union: $A \cup B = B \cup A$
  • No matter the order, the result is the same.
• Intersection: $A \cap B = B \cap A$
  • No matter the order, the result is the same. Imagine: Whether you add the apples to the basket first or the oranges, you'll end up with the same basket of fruit.

Associative Property

• Union: $(A \cup B) \cup C = A \cup (B \cup C)$
  • Grouping doesn't change the outcome.
• Intersection: $(A \cap B) \cap C = A \cap (B \cap C)$
  • Grouping doesn't change the outcome. Imagine: Whether you group the apple-lovers and the banana-lovers first, or the banana-lovers and the cherry-lovers, it doesn't change the total group of fruit-lovers.

Putting It All Together

Let's look at an example where we combine everything we've learned:

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