Q.State whether the following sets are equal or not: $I = \{x : x^2 - 5x + 7 = 0,\ x \in R\}$ and $J = \phi$

Null Set Identification: Understanding the "Why" Behind the Definition

What is a Null Set?

A null set (also called an empty set) is a set that contains no elements. It is denoted by the symbol $\emptyset$ or $\{ \}$.

The key idea is not a "formula" in the traditional sense, but a definitional truth that flows from the very nature of what a set is.

The Core Principle: Why $\emptyset$ Exists

The Axiom of Empty Set (Zermelo-Fraenkel Set Theory)

In formal set theory, the existence of the null set is an axiom — a starting assumption we accept without proof. But we can understand why it must be true:

Intuition: If you can have a set of apples, you can also have a set of nothing. The concept of "collection" does not require that the collection actually contain anything.

The Defining Property

The null set is the unique set that satisfies:

This reads: "For every object $x$, $x$ is not an element of the empty set."

Why this holds:

• If there were any element in $\emptyset$, it would violate the definition.
• The statement is vacuously true — there is no counterexample because there is nothing to check.

Key "Formulae" and Their Reasoning

1. Cardinality: $|\emptyset| = 0$

Why:

The cardinality of a set is the number of distinct elements it contains. Since $\emptyset$ has no elements, the count is zero.

This is not a derived result — it is the definition of zero applied to set size.

2. Subset Relationship: $\emptyset \subseteq A$ for any set $A$

Why this is always true:

The definition of subset: $A \subseteq B$ means "every element of $A$ is also an element of $B$."

For $\emptyset \subseteq A$:

• We need to check: "Is every element of $\emptyset$ also in $A$?"
• Since $\emptyset$ has no elements, there is nothing to check.
• The statement is vacuously true — there is no counterexample.

Key insight: The empty set is a subset of every set, including itself.

3. Union with Null Set: $A \cup \emptyset = A$

Why:

The union of two sets contains all elements that belong to at least one of them.

Since $\emptyset$ contributes no new elements, the result is exactly $A$.

Derivation:

• If $x \in A$, then $x \in A \cup \emptyset$ (by definition of union).
• If $x \notin A$, then $x \notin \emptyset$ (since nothing is in $\emptyset$), so $x \notin A \cup \emptyset$.
• Therefore, $A \cup \emptyset$ contains exactly the elements of $A$.

4. Intersection with Null Set: $A \cap \emptyset = \emptyset$

Why:

The intersection of two sets contains elements that belong to both sets.

Since $\emptyset$ has no elements, the condition "$x \in \emptyset$" is never true. Therefore, no element can satisfy both conditions.

Result: The intersection is empty — it is the null set itself.

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