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

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In summary, we can prove that a-(b∩ c)=(a-b)u(a-c) using the rules of set manipulation, specifically DeMorgan's law and the distributive property. By substituting the definitions of set subtraction and intersection, we can show that the left-hand side is equivalent to the right-hand side. This can also be illustrated using a Venn diagram.
  • #1
Princess Shaina
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Prove that
a-(b∩ c)=(a-b)u(a-c)
 
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  • #2
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)
 
  • #3
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
 
  • #4
Have you tried using the Venn diagram?
 
  • #5
Princess Shaina said:
Prove that
a-(b∩ c)=(a-b)u(a-c)

[tex]\begin{array}{ccccc} 1. & a - (b \cap c) && 1. & \text{LHS} \\
2. & a \cap \overline{(b \cap c)} && 2. & \text{Def. subtr'n} \\
3. & a \cap ( \overline b \cup \overline c) & & 3. & \text{DeMorgan} \\
4. & (a \cap{\overline b}) \cup (a \cap {\overline c}) && 4. & \text{Distributive} \\
5. & (a - b) \cup (a - c) && 5. & \text{Def. subtr'n} \\
&&&& \text{RHS}\end{array}
[/tex]

 

1. What is the meaning of "A minus B Intersect C"?

A minus B Intersect C refers to the set of elements that are common to both A minus B and C. In other words, it is the intersection of the two sets A minus B and C.

2. How is "A minus B Union A minus C" different from "A minus B Intersect C"?

The set A minus B Union A minus C includes all the elements that are present in either A minus B or A minus C, or both. On the other hand, A minus B Intersect C only includes the elements that are common to both A minus B and C.

3. Can you prove that A minus B Intersect C equals A minus B Union A minus C?

Yes, the equality can be proven using the distributive property of set operations. A minus B Intersect C can be rewritten as (A minus B) Intersect (A minus C), which is equal to (A minus B) Union (A minus C) using the distributive property. Thus, A minus B Intersect C equals A minus B Union A minus C.

4. What is the significance of proving A minus B Intersect C equals A minus B Union A minus C?

Proving this equality is important in set theory because it helps us understand the relationship between set operations. It also allows us to use different operations interchangeably in certain situations, making it easier to solve problems and prove other theorems.

5. Are there any real-life applications of this concept?

Yes, this concept is commonly used in computer science and data analysis to perform operations on sets of data. For example, in database querying, the intersection of two sets can be used to retrieve data that satisfies multiple conditions, while the union of two sets can be used to retrieve data that satisfies either of the conditions. This concept is also used in everyday life, such as when organizing and categorizing items based on their shared characteristics.

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