Decide which of the 5 properties of a commutative group (closure, identity, inverse, associative, commutative) hold for
the given system.
-$$\begin{array} { c | c c c c }
‡ & 4 & 9 & 14 & 19 \
\hline 4 & 14 & 19 & 4 & 9 \
9 & 19 & 4 & 9 & 14 \
14 & 4 & 9 & 14 & 19 \
19 & 9 & 14 & 19 & 4
\end{array}$$
A) Closure, identity, inverse, and commutative only
B) Closure, identity, associative, and commutative only
C) Closure, identity, inverse, and associative only
D) All of them

Question

Decide which of the 5 properties of a commutative group (closure, identity, inverse, associative, commutative) hold for
the given system.
-$$\begin{array} { c | c c c c } 
‡ & 4 & 9 & 14 & 19 \ 
\hline 4 & 14 & 19 & 4 & 9 \
9 & 19 & 4 & 9 & 14 \
14 & 4 & 9 & 14 & 19 \
19 & 9 & 14 & 19 & 4
\end{array}$$
A) Closure, identity, inverse, and commutative only
B) Closure, identity, associative, and commutative only
C) Closure, identity, inverse, and associative only
D) All of them

Quizplus · Accepted Answer

The given system is closed because every operation within the set results in an element from the set. There is an identity element (14), as operating it with any element returns that element. Each element has an inverse (operating any two distinct non-identity elements results in the third one, satisfying the inverse property). The operation is associative because the operation's result does not depend on the grouping of the elements (this can be verified through the table, though it's tedious). Lastly, the operation is commutative because the table is symmetric along the diagonal, indicating that the order of the operation does not affect the outcome.

Decide Which of the 5 Properties of a Commutative Group