Decide which of the 5 properties of a commutative group (closure, identity, inverse, associative, commutative) hold for
the given system.
-$$\begin{array} { c | c c c c }
/ & 5 & 11 & 16 & 8 \
\hline 5 & 11 & 16 & 8 & 5 \
11 & 16 & 8 & 5 & 11 \
16 & 8 & 5 & 11 & 16 \
8 & 5 & 11 & 16 & 8
\end{array}$$
A) Closure, identity, associative, and commutative only
B) All of them
C) Closure, identity, inverse, and commutative only
D) Closure, identity, inverse, and associative only

Question

Decide which of the 5 properties of a commutative group (closure, identity, inverse, associative, commutative) hold for
the given system.
-$$\begin{array} { c | c c c c } 
/ & 5 & 11 & 16 & 8 \
\hline 5 & 11 & 16 & 8 & 5 \
11 & 16 & 8 & 5 & 11 \
16 & 8 & 5 & 11 & 16 \
8 & 5 & 11 & 16 & 8
\end{array}$$
A) Closure, identity, associative, and commutative only
B) All of them
C) Closure, identity, inverse, and commutative only
D) Closure, identity, inverse, and associative only

Quizplus · Accepted Answer

All properties hold for this system. Closure is evident as all operations within the set result in elements that are also within the set. The identity element is 5, as operating it with any element returns that element. Each element has an inverse (5 with 5, 11 with 11, etc.), ensuring every element can be "undone" to return to the identity element. The table is symmetric relative to the diagonal, indicating commutativity. Associativity, while not directly verifiable from the table without specific calculations, is assumed to hold given the context of the question and the properties of a commutative group.

Decide Which of the 5 Properties of a Commutative Group