Math Problem Statement
Solution
The statement provided is:
"For two finite sets and : "
Where:
- means that sets and are not equal.
- refers to the cardinality (number of elements) of the intersection of and .
- is the cardinality of set .
Analysis:
The statement claims that if and are different sets, then the number of elements in (the intersection) must be strictly less than the number of elements in .
However, this is not necessarily true. Consider the case where:
- , meaning all elements of are contained in , but has additional elements not in . In this case, , but , so the cardinality would equal , 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 , even if .
Thus, the selected answer "False" is correct.
Would you like more details or have any further questions?
Here are 5 related questions:
- What is the definition of set equality?
- How can we find the intersection of two sets?
- Can the intersection of two sets ever be larger than one of the sets?
- What happens when one set is a subset of the other?
- 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