Math Problem Statement

For two finite sets A, B: A ≠ B ⇒ #(A ∩ B) < #B

Solution

The statement provided is:

"For two finite sets AA and BB: AB    #(AB)<#BA \neq B \implies \#(A \cap B) < \#B "

Where:

  • ABA \neq B means that sets AA and BB are not equal.
  • #(AB)\#(A \cap B) refers to the cardinality (number of elements) of the intersection of AA and BB.
  • #B\#B is the cardinality of set BB.

Analysis:

The statement claims that if AA and BB are different sets, then the number of elements in ABA \cap B (the intersection) must be strictly less than the number of elements in BB.

However, this is not necessarily true. Consider the case where:

  • BAB \subseteq A, meaning all elements of BB are contained in AA, but AA has additional elements not in BB. In this case, ABA \neq B, but AB=BA \cap B = B, so the cardinality #(AB)\#(A \cap B) would equal #B\#B, not be less than it.

This provides a counterexample, showing the statement is false. The cardinality of the intersection is not always strictly less than that of BB, even if ABA \neq B.

Thus, the selected answer "False" is correct.

Would you like more details or have any further questions?

Here are 5 related questions:

  1. What is the definition of set equality?
  2. How can we find the intersection of two sets?
  3. Can the intersection of two sets ever be larger than one of the sets?
  4. What happens when one set is a subset of the other?
  5. What is the union of two sets, and how does it differ from the intersection?

Tip: When proving or disproving set relations, always look for counterexamples to test the general validity of the statement.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Cardinality
Intersection of Sets

Formulas

#(A ∩ B)
#B

Theorems

Cardinality of Sets
Set Subset Properties

Suitable Grade Level

Grades 9-12