How can one "cancel out" set union as if I am using the additive property of equality of real numbers?

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I am aware that for equalities involving, for instance involving real numbers, we have the cancellation property, like in the following:

$\textit{x} \text{ +} \text{ 3} = \text{5}$,

We get the value of $\textit{x}$ by subtracting 3 from both sides of the equation to get $\textit{x} = \text{2}$.

Now suppose I have the following statement,

$\textit{A} \cup \textit{C} = \textit{B} \cup \textit{C}$

I want to show that A = B. Can I safely "cancel out" $\cup \textit{ C}$ from both sides of the equation, and if so, how?

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1 Answer

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No you cannot cancel out C.

Suppose $C = \{1,2,3,4,5\}, A = \{1,2\}, B = \emptyset$

Then $A \cup C = B \cup C$ but $A \neq B$.

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