Math Problem Statement

true or false

If A ⊆ B, then n(A) ≤ n(B).

Solution

The statement in the image reads:

"If ABA \subseteq B, then n(A)n(B)n(A) \leq n(B)."

This statement is true. Here’s why:

  1. Subset Definition: If AA is a subset of BB (ABA \subseteq B), every element of AA is also an element of BB.
  2. Cardinality: The notation n(A)n(A) represents the cardinality (or number of elements) of set AA.
  3. Implication: Since AA is a subset of BB, the number of elements in AA cannot exceed the number of elements in BB. This means that n(A)n(B)n(A) \leq n(B).

Therefore, the statement is true.

Would you like further details or have any questions?

Here are some related questions to expand your understanding:

  1. What is the definition of a proper subset?
  2. How does the cardinality of the union of two sets AA and BB relate to the cardinalities of AA and BB?
  3. What happens to the cardinality relationship if AA is a proper subset of BB (ABA \subset B)?
  4. Can you provide an example where ABA \subseteq B and illustrate n(A)n(B)n(A) \leq n(B) with specific numbers?
  5. How does the inclusion-exclusion principle apply to the cardinalities of sets?

Tip: When dealing with set operations and cardinality, visualizing with Venn diagrams can often help clarify relationships and make understanding easier.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Subsets
Cardinality

Formulas

n(A) ≤ n(B)

Theorems

Subset Definition
Cardinality Theorem

Suitable Grade Level

Grades 9-12