Prove A minus B Intersect C Equals A minus B Union A minus C

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Discussion Overview

The discussion centers around proving the set equality \( A - (B \cap C) = (A - B) \cup (A - C) \). Participants explore various methods of proof, including set manipulation rules and visual aids like Venn diagrams.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose using set manipulation rules, such as DeMorgan's laws and distributive properties, to prove the equality.
  • One participant provides a step-by-step breakdown of the proof using definitions of set subtraction and properties of intersections and unions.
  • Another participant suggests the use of Venn diagrams as a potential method for visualizing the proof.
  • There are repeated requests for clarification on the rules of set manipulation that can be applied in this context.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the proof method, with multiple approaches being discussed and no clear resolution on which is preferred or most effective.

Contextual Notes

Some limitations include potential missing assumptions regarding the definitions of set operations and the applicability of certain rules in specific contexts.

Princess Shaina
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Prove that
a-(b∩ c)=(a-b)u(a-c)
 
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Princess Shaina said:
Prove that
a-(b∩ c)=(a-b)u(a-c)

Hello Your Highness! Welcome to MHB! ;)

Which rules for set manipulation can we use? (Wondering)
 
A-(B⋂C)=(A-B)⋃(A-C) If A-B={xlx∈A and x∉B} A-C={xlx∈A and x∉C} then (A-B)⋃(A-C)={xlx∈A, x∉(B and C) Let X=A and Y=(B⋂C) X-Y={xlx∈X and x∉Y} x∉Y x∉(B⋂C) x∉(B and C) physicsforums
 
Have you tried using the Venn diagram?
 
Princess Shaina said:
Prove that
a-(b∩ c)=(a-b)u(a-c)
\begin{array}{ccccc} 1. &amp; a - (b \cap c) &amp;&amp; 1. &amp; \text{LHS} \\ <br /> 2. &amp; a \cap \overline{(b \cap c)} &amp;&amp; 2. &amp; \text{Def. subtr&#039;n} \\ <br /> 3. &amp; a \cap ( \overline b \cup \overline c) &amp; &amp; 3. &amp; \text{DeMorgan} \\<br /> 4. &amp; (a \cap{\overline b}) \cup (a \cap {\overline c}) &amp;&amp; 4. &amp; \text{Distributive} \\<br /> 5. &amp; (a - b) \cup (a - c) &amp;&amp; 5. &amp; \text{Def. subtr&#039;n} \\<br /> &amp;&amp;&amp;&amp; \text{RHS}\end{array}<br />

 

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