Question

Prove that if $A$ and $B$ are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$.

   Prove that if $A$ and $B$ are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$.

Mathematical Proofs: A Transition to Advanced Mathematics

Mathematical Proofs: A Transition to Advanced Mathematics

Gary Chartrand,… 3rd Edition

Chapter 4, Problem 44

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We need to prove that if the union of two sets \(A\) and \(B\) is not empty, then at least one of the sets \(A\) or \(B\) is not empty. Show more…

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Prove that if $A$ and $B$ are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$.

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Key Concepts

Union of Sets

The union A ? B is defined as the set of elements that are in A, or in B, or in both. This concept is fundamental in set theory and involves the idea that membership in the union comes from membership in at least one of the contributing sets.

Empty Set

The empty set is the unique set that contains no elements. Its important property in this context is that if both sets A and B are empty, then their union is empty, making it central to understanding conditions for non-emptiness.

Logical Disjunction

Logical disjunction, often expressed as 'or', means that if at least one of the conditions holds, the entire statement is true. This is directly related to the idea that the union of two sets is non-empty if at least one of them contains an element.

Proof by Contrapositive

Proof by contrapositive involves proving an implication by demonstrating that the negation of the conclusion implies the negation of the premise. In this case, showing that if both A and B are empty then A ? B is empty, which supports the original claim.

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