Math Problem Statement

true or false

A ⊆ B ∩ C ⇒ P(A) = P(B) ∩ C. Is this statement true or false?

Solution

The statement in the image appears to be a logical or mathematical proposition involving subsets and power sets:

The expression is: ABC    P(A)=P(B)CA \subseteq B \cap C \implies P(A) = P(B) \cap C

Analysis:

  1. Understanding the expression:

    • ABCA \subseteq B \cap C: AA is a subset of the intersection of BB and CC, meaning every element of AA is in both BB and CC.
    • P(A)P(A): The power set of AA, which is the set of all subsets of AA.
    • P(B)CP(B) \cap C: Intersection of the power set of BB with CC. However, this notation is non-standard because P(B)P(B) (a set of subsets) and CC (a set, not necessarily related to subsets) are not directly comparable. If CC is intended to refer to subsets, then the meaning must be clarified.
  2. Logical interpretation:

    • For the equality P(A)=P(B)CP(A) = P(B) \cap C, P(B)CP(B) \cap C should involve subsets of CC that also belong to P(B)P(B).
  3. True or False?

    • If ABCA \subseteq B \cap C, then AA is a subset of both BB and CC. Hence, P(A)P(A) is a subset of both P(B)P(B) and P(C)P(C).
    • However, P(B)CP(B) \cap C does not clearly align because CC and P(B)P(B) operate on different levels unless additional context specifies that ( C \

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Math Problem Analysis

Mathematical Concepts

Set Theory
Subsets
Power Sets
Logical Implication

Formulas

Power Set: P(A)
Set Intersection: B ∩ C

Theorems

Subset Definition: A ⊆ B if every element of A is also in B

Suitable Grade Level

University-level or Advanced High School