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

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SUMMARY

The proof that A minus (B intersect C) equals (A minus B) union (A minus C) is established through set manipulation rules. Starting with the left-hand side, A - (B ∩ C) is rewritten using DeMorgan's laws and distributive properties to arrive at the right-hand side. The steps include defining set subtraction, applying DeMorgan's theorem, and utilizing the distributive property to confirm the equality. This proof is essential for understanding basic set theory operations.

PREREQUISITES
  • Understanding of set theory concepts, including unions and intersections.
  • Familiarity with DeMorgan's laws in set operations.
  • Knowledge of set subtraction and its definition.
  • Ability to manipulate set expressions using distributive properties.
NEXT STEPS
  • Study advanced set theory concepts, including cardinality and power sets.
  • Learn about Venn diagrams and their applications in visualizing set operations.
  • Explore proofs in set theory to strengthen logical reasoning skills.
  • Investigate applications of set theory in computer science, particularly in database management.
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Mathematicians, educators, students studying set theory, and anyone interested in logical proofs and set operations.

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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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