> [!summary] A subset is a collection of elements from a larger set >[!info]+ Read Time **⏱ 1 min** # Definition A subset is a smaller collection of things taken from a larger [[Sets|set]]. If $A$ and $B$ are two sets. The set $A$ is only a subset of $B$ if every element of $A$ is also an element of $B$, formally defined below. $ A \subseteq B \quad \text{If any only if} \space \forall x\in A, x\in B $ # Examples > [!example] Are these set $A$ a subset of $B$ If $A =\left\{ 1,2,3 \right\}$ and $B = \left\{ 1,2,3,4,5,6\right\}$ Every element of A is an element of B, so by definition $A \subseteq B$ > [!example] Is the [[Empty Set|empty set]] a subset of B If $A = \emptyset$ and $B = \left\{ 1,2,3,4,5\right\}$ > By definition of a subset, every element of A must be an element of B. There are no elements in A to check if it's an element of B, so by the definition is satisfied and $A \subseteq B$ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/-4KLugkLwHQ?si=HXucfLqTsEJaCHpE" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>