Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 2 - Section 2.5 - Cardinality of Sets - Exercises - Page 176: 12

Answer

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Work Step by Step

(If there is a one-one function from $A$ to $B$, then $|A|\leq|B|$ Given: $A$ and $B$ are sets with $A\subset B$ To prove: $|A|\leq|B|$ Proof: By the definition of a subset: If $a \in A$, then $a \in B$ We can then define the function f as: $$f: A\implies B, f(a) = a$$ We need to check that the function f is one-to-one. Let $f(a)= f(b)$. By the definition of f , we then obtain $a=b$. Thus f is one-to-one. Since f is a one-to-one function from $A$ to $B$ , $|A|\leq |B|$
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