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 />

 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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