Subsets

The Idea of a Subset

When you have two sets, it is natural to ask whether one fits entirely inside the other. Consider the set of all students in your school — call it $X$ — and the set of all students in your class — call it $Y$. Every student in your class is also a student in your school. In other words, every element of $Y$ is also an element of $X$. When this happens, we say that $Y$ is a subset of $X$.

The symbol for "is a subset of" is $\subset$. So we write $Y \subset X$, which reads as "$Y$ is a subset of $X$" or "$Y$ is contained in $X$".

Formal Definition of a Subset

Definition. A set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of $B$.

In symbols, we write $A \subset B$ if, whenever $a \in A$, then $a \in B$. The symbol $\Rightarrow$ means "implies". Using it, the definition becomes:

We read this as: "$A$ is a subset of $B$ if $a$ being an element of $A$ implies that $a$ is also an element of $B$."

If $A$ is not a subset of $B$, we write $A \not\subset B$.

When Two Sets Are Equal

If it happens that every element of $B$ is also in $A$, then we have $B \subset A$ as well. When both $A \subset B$ and $B \subset A$ hold, the two sets are exactly the same. This gives us a powerful way to define set equality:

The symbol $\iff$ stands for "if and only if" (often written as "iff"). It means the implication works in both directions: if $A = B$, then certainly $A \subset B$ and $B \subset A$; conversely, if both subset relations hold, the sets must be equal.

Every Set Is a Subset of Itself

From the definition, it follows immediately that every set $A$ is a subset of itself. Why? Because every element of $A$ is obviously an element of $A$. So $A \subset A$ is always true.

The Empty Set Is a Subset of Every Set

The empty set $\phi$ has no elements. The condition "$a \in \phi \Rightarrow a \in B$" is vacuously true — there is no $a$ to violate it. By convention and logical necessity, we agree that:

Examples to Illustrate Subsets

(i) The set $\mathbb{Q}$ of rational numbers is a subset of the set $\mathbb{R}$ of real numbers. We write $\mathbb{Q} \subset \mathbb{R}$.

(ii) Let $A$ be the set of all divisors of 56, and $B$ the set of all prime divisors of 56. Every prime divisor is a divisor, so $B \subset A$.

(iii) Let $A = \{1, 3, 5\}$ and $B = \{x : x \text{ is an odd natural number less than } 6\}$. The odd natural numbers less than 6 are 1, 3, and 5. So $B = \{1, 3, 5\}$. Here $A \subset B$ and $B \subset A$, hence $A = B$. …