Exercise · ZW-ZERMELO-FRAENKEL-AXIOMS-005 · ★★☆ Intermediate
If \( A=\{1,2\} \) and \( B=\{3\} \), how many elements does \( A\times B \) have?
Choose one option.
Show answer
- A. 2
Explanation
The pairs are (1,3) and (2,3), so there are two elements. Adding cardinalities instead of multiplying gives a wrong result.
Review the concept: Zermelo–Fraenkel axioms
The Zermelo-Fraenkel axioms (ZF) are the standard axiom system of set theory, formulated in first-order logic. They allow the usual mathematics to be built from the primitive notions of set and membership (\( \in \)).
\[\forall x\,\forall y\,(\forall z\,(z\in x \leftrightarrow z\in y)\rightarrow x=y)\]
More practice problems: zermelo–fraenkel axioms
- ZW-ZERMELO-FRAENKEL-AXIOMS-001 Which of these statements is the axiom of extensionality?
- ZW-ZERMELO-FRAENKEL-AXIOMS-002 What is the name of the axiom that guarantees the existence of \a,b\ for any a and b ?
- ZW-ZERMELO-FRAENKEL-AXIOMS-003 Which axiom guarantees the existence of a set containing the empty set that is closed under success…
- ZW-ZERMELO-FRAENKEL-AXIOMS-004 Which axiom allows forming the set of all subsets of a given set?
- ZW-ZERMELO-FRAENKEL-AXIOMS-006 What is the union of the family \\1\,\1,2\,\2,3\\ ?
- ZW-ZERMELO-FRAENKEL-AXIOMS-007 Which of these statements about regularity is correct?
- ZW-ZERMELO-FRAENKEL-AXIOMS-008 Which axiom allows defining an image set by applying a formula to a given set?
- ZW-ZERMELO-FRAENKEL-AXIOMS-009 Let A=\∅,\∅\\ . How many elements does P(A) have, and what is correct about A ∈ A under regularity?
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