Revisiting the Null Set — A Subset of Every Set

You already know from the previous chapter that a set $A$ is a subset of a set $B$ (written $A \subseteq B$) whenever every element of $A$ is also an element of $B$. The empty set $\emptyset$ deserves special attention here, because a genuinely important — and, at first glance, slightly strange-looking — fact follows directly from this definition.

Why should this be true, even when $A$ itself has nothing to do with $\emptyset$? Go back to the definition: $\emptyset \subseteq A$ means "every element of $\emptyset$ is also an element of $A$." But $\emptyset$ has no elements at all — so there is nothing to check. The statement "every element of $\emptyset$ is in $A$" is true simply because there is no element of $\emptyset$ that could ever fail the test. A statement that is true only because its condition is never actually triggered is called vacuously true — this is exactly that situation, and it is why the theorem holds for every set $A$ without exception, including when $A$ itself is $\emptyset$ (then $\emptyset \subseteq \emptyset$, which is also true, since every set is a subset of itself).

This chapter takes the vocabulary built in the previous chapter — sets, subsets, cardinality, Cartesian products — and puts it to work: combining sets with operations (union, intersection, difference, symmetric difference, complement), picturing those operations with Venn diagrams, counting elements across combined sets with the inclusion-exclusion principle, and finally using sets of ordered pairs to picture functions. The fact that $\emptyset \subseteq A$ for every set $A$ resurfaces almost immediately in the very next section, once we start listing every possible subset of a set — a skill this WBCHSE Class 11 Business Mathematics and Basic Statistics chapter builds from the ground up.