Math Problem Statement

  1. Give a general prove of the statement below. Let X and Y be two non-empty sets. If Y ⊆ X, then show that X ∪ Y = X.

Solution

To prove the statement "If YXY \subseteq X, then XY=XX \cup Y = X" for any two non-empty sets XX and YY, we will use the definitions of set union and subset.

Definitions:

  1. Union of Sets: The union of two sets XX and YY, denoted XYX \cup Y, is the set of all elements that are in XX, in YY, or in both. XY={zzX or zY}X \cup Y = \{ z \mid z \in X \text{ or } z \in Y \}
  2. Subset: A set YY is a subset of a set XX, denoted YXY \subseteq X, if every element of YY is also an element of XX. YX    (yY,yX)Y \subseteq X \iff (\forall y \in Y, y \in X)

Proof:

To prove that XY=XX \cup Y = X when YXY \subseteq X, we need to show two inclusions:

  1. XYXX \cup Y \subseteq X
  2. XXYX \subseteq X \cup Y

1. Prove XYXX \cup Y \subseteq X:

Let zz be an arbitrary element of XYX \cup Y. By the definition of union: zXY    zX or zYz \in X \cup Y \implies z \in X \text{ or } z \in Y We need to show that zXz \in X.

  • If zXz \in X, then zXz \in X.
  • If zYz \in Y, since YXY \subseteq X, zXz \in X.

In both cases, we have zXz \in X. Therefore, zXY    zXz \in X \cup Y \implies z \in X Hence, XYXX \cup Y \subseteq X.

2. Prove XXYX \subseteq X \cup Y:

Let zz be an arbitrary element of XX. By the definition of union: zX    zXYz \in X \implies z \in X \cup Y because zXz \in X satisfies the condition for being in the union XYX \cup Y.

Thus, zX    zXYz \in X \implies z \in X \cup Y Hence, XXYX \subseteq X \cup Y.

Conclusion:

Since we have shown both XYXX \cup Y \subseteq X and XXYX \subseteq X \cup Y, it follows that: XY=XX \cup Y = X

This completes the proof.


Would you like more details on any part of this proof or have any questions?

Here are some related questions you might find interesting:

  1. How do you prove that XY=YX \cap Y = Y if YXY \subseteq X?
  2. What is an example where Y⊈XY \not\subseteq X and XYXX \cup Y \neq X?
  3. How do you define the intersection of two sets?
  4. Can the empty set be a subset of any set? Why?
  5. How would you prove that the union of a set with the empty set is the set itself?
  6. What is the symmetric difference of two sets and how is it defined?
  7. How do you prove that X(XY)=XX \cup (X \cap Y) = X?
  8. What are the properties of set union and intersection that are used in proofs?

Tip: When working with sets, always start by carefully defining the sets and the operations you are using. This helps in constructing clear and rigorous proofs.

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Math Problem Analysis

Mathematical Concepts

Set Theory

Formulas

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Theorems

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Suitable Grade Level

Advanced