subset

let $\text{[math]}$ be sets. we say that $\text{[math]}$ is a subset of $\text{[math]}$ , denoted $\text{[math]}$ , iff every element of $\text{[math]}$ is also an element of $\text{[math]}$ , i.e.,
$\text{[math]}$ .
[cite:@tao_analysis_1 section 3 set theory]

we say that $\text{[math]}$ is a proper subset of $\text{[math]}$ , denoted $\text{[math]}$ , if $\text{[math]}$ and $\text{[math]}$ .
[cite:@tao_analysis_1 section 3 set theory]

element chasing is showing a set is a subset of another by showing that all its elements are also in the other set.