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

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The discussion focuses on proving the set identity a - (b ∩ c) = (a - b) ∪ (a - c). Participants explore the rules of set manipulation, including definitions of set subtraction and De Morgan's laws. The proof is structured using logical steps that transform the left-hand side into the right-hand side through distributive properties and set definitions. A Venn diagram is suggested as a visual aid for understanding the relationship between the sets. The conversation emphasizes the importance of clear definitions and logical reasoning in set theory proofs.
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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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