Math Challenges

Submissions for Problem #8

Problem #8

Recall that a binary operation on a set S takes two elements of set S and maps them to an element of set S. Are the following considered binary operations?

\[ \displaylines{1)S=\mathbb{Z},a\ast b=a^4b^3\\ 2)S=\mathbb{Z},a\ast b=\frac{a^2}{b}} \]

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fish-face

Solution: \( \displaylines{\text{yes}\\ \text{no}} \)

Explanation:
1. integers are closed under multiplication and hence also exponentiation 2. if a=1, b=2, a*b = 1/2 which is not an integer.

zapwai

Solution: \( \displaylines{a^4b^3\in\mathbb{Z}\text{ if }a,b\in\mathbb{Z}\\ \\ \text{If }a=2,b=3,\text{ then }\frac{a^2}{b}=\frac43\notin\mathbb{Z}} \)

Explanation:
The integers are closed under multiplication, but not division. Powers and products like those in part (1) will still be integers. However the division in part (2) will fail to be an integer as soon as a is not divisible by b.