How do I calculate the number of elements in set Z using set theory?

In summary, the conversation discusses the calculation of the number of elements in a set Z that consists of all elements in either set X or set Y, except for the k elements that are common to both sets. The answer is x + y - 2k, which can be obtained by using the formula X sym diff Y = (X \ Y) u (Y \ X) and understanding that the union of disjoint sets has a cardinality equal to the sum of the cardinalities of the individual sets. It is also mentioned that proof by example is not a valid proof.
  • #1
courtrigrad
1,236
2
Hello all

Set X has x elements and Sset Y has y elements and Set Z consists of all elements are are in either set X or set Y with the exception of the k common elements. I know that the answer is

x+ y - 2k, however how would I get this? Should I just use a practical example?
 
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  • #2
No, proof by example isn't a proof.

X sym diff Y = (X \ Y)u(Y\X)

the union is of disjoint sets so the card of the lhs is the sum of the cards on the rhs. now just find card (X\Y) hint X\Y = X\(XnY)
 
  • #3
You are told that there are k elements in both X and Y. That means there are x-k elements in x that are NOT in y and y- k elements that are in Y but NOT in X. Z will contain (x- k)+ (y- k)= x+ y- 2k elements.
 

FAQ: How do I calculate the number of elements in set Z using set theory?

What is Set Theory and why is it important?

Set Theory is a branch of mathematics that deals with the study of sets, which are collections of objects or elements. It is important because it provides a foundation for other areas of mathematics and has applications in various fields such as computer science, statistics, and logic.

What are the basic concepts in Set Theory?

The basic concepts in Set Theory include sets, elements, subsets, unions, intersections, and complements. Sets are collections of objects or elements, while elements are the individual objects that make up a set. Subsets are sets that contain elements from a larger set, and unions and intersections are operations that combine or compare sets. Complements are sets that contain all the elements that are not in a given set.

What is the difference between a finite and infinite set?

A finite set is a set that has a specific number of elements, while an infinite set has an uncountable or unlimited number of elements. For example, the set of numbers 1, 2, and 3 is a finite set, while the set of all natural numbers is an infinite set.

How is Set Theory related to logic?

Set Theory and logic are closely related because they both deal with the concept of collections and their relationships. Set Theory provides a formal language and rules for defining and manipulating sets, while logic provides a framework for reasoning and making inferences about these sets.

Can Set Theory be applied in real-life situations?

Yes, Set Theory has many practical applications in fields such as computer science, statistics, and economics. For example, it is used in database management to organize and retrieve data, in probability to calculate the likelihood of events, and in decision-making processes to analyze and compare different options.

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