Set Difference Intersection with Second Set is Empty Set
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Theorem
Let $S$ and $T$ be sets.
The intersection of the set difference of $S$ and $T$ with $T$ is the empty set:
- $\paren {S \setminus T} \cap T = \O$
Proof 1
\(\ds \paren {S \setminus T} \cap T\) | \(=\) | \(\ds \paren {S \cap T} \setminus \paren {T \cap T}\) | Set Intersection Distributes over Set Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {S \cap T} \setminus T\) | Set Intersection is Idempotent | |||||||||||
\(\ds \) | \(=\) | \(\ds \O\) | Set Difference of Intersection with Set is Empty Set |
$\blacksquare$
Proof 2
\(\ds \paren {S \setminus T} \cap T\) | \(=\) | \(\ds \paren {S \cap T} \setminus T\) | Intersection with Set Difference is Set Difference with Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds \O\) | Set Difference of Intersection with Set is Empty Set |
$\blacksquare$
Sources
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.1$: Theorem $1.5$
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.2$: Operations on Sets: Exercise $1.2.5 \ \text{(iii)}$