Set Intersection is not Cancellable
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Theorem
Set intersection is not a cancellable operation.
That is, for a given $A, B, C \subseteq S$ for some $S$, it is not always the case that:
- $A \cap B = A \cap C \implies B = C$
Proof
Let $S = \set {a, b, c}$.
Let:
- $A = \set a$
- $B = \set {a, b}$
- $C = \set {a, c}$
Then:
\(\ds A \cap B\) | \(=\) | \(\ds \set a\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds A \cap C\) |
but:
\(\ds B\) | \(=\) | \(\ds \set {a, b}\) | ||||||||||||
\(\ds \) | \(\ne\) | \(\ds \set {a, c}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds C\) |
$\blacksquare$
Sources
- 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.1$: Sets: Exercise $\text{B} \ 3$