Correctly define each of the following using formal logic and ∈: 1. x ∈ A - B 2. A ⊆ B
Understand the Problem
The question is asking for formal definitions of two mathematical expressions involving set notation: 'x ∈ A - B' and 'A ⊆ B'. The goal is to define them using formal logic and the element relation '∈'.
Answer
1. \( x \in A - B \iff (x \in A) \land (x \notin B) \) 2. \( A \subseteq B \iff \forall x ( x \in A \implies x \in B ) \)
Answer for screen readers
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( x \in A - B \iff (x \in A) \land (x \notin B) )
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( A \subseteq B \iff \forall x ( x \in A \implies x \in B ) )
Steps to Solve
- Define the expression ( x \in A - B )
The expression ( x \in A - B ) means that ( x ) is an element of the set difference between sets ( A ) and ( B ). Formally, it can be defined as: $$ x \in A - B \iff (x \in A) \land (x \notin B) $$
- Define the expression ( A \subseteq B )
The expression ( A \subseteq B ) indicates that set ( A ) is a subset of set ( B ). This can be defined formally as: $$ A \subseteq B \iff \forall x ( x \in A \implies x \in B ) $$
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( x \in A - B \iff (x \in A) \land (x \notin B) )
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( A \subseteq B \iff \forall x ( x \in A \implies x \in B ) )
More Information
The definitions provided are fundamental concepts in set theory. The first definition describes how to determine if an element belongs to the set difference, while the second definition explains what it means for one set to be a subset of another. Understanding these definitions is crucial for deeper studies in mathematics, particularly in areas involving functions, relations, and proofs.
Tips
- Misunderstanding the notation of set difference: Remember that ( A - B ) is not the same as ( A \cap B ). ( A - B ) includes elements that are in ( A ) but not in ( B ).
- Confusing subset notation ( A \subseteq B ) with strict inequality ( A \subset B ): The former includes the possibility of ( A ) being equal to ( B ), while the latter does not.
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