Binary operations $*$, $\oplus$, $\odot$ defined on the set of rational numbers
I. $a * b = a - b$ II. $a \oplus b = a + b + ab$ III. $a \odot b = \frac{a+b}{5}$
are defined as follows. Accordingly, which of these operations satisfy the associative property?
A) Only I
B) Only II
C) Only III
D) I and II
E) II and III

On the set $A = \{1,2,3,4,5\}$ $$f = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 1 & 5 & 2 & 4 \end{pmatrix}, \quad g = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 3 & 4 & 1 & 2 \end{pmatrix}$$ For the permutations, what is the value of $g f^{-1}(2)$?
A) 1
B) 2
C) 3
D) 4
E) 5

The $\Delta$ operation on the set $\mathrm { A } = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e } \}$ is defined by the table below. For example, a $\Delta \mathrm { d } = \mathrm { c }$ and $\mathrm { d } \Delta \mathrm { a } = \mathrm { a }$.

$\Delta$	a	b	c	d	e
a	a	b	a	c	d
b	c	b	b	a	e
c	a	b	c	d	e
d	a	a	d	d	b
e	e	e	e	d	a

According to this table, which of the following subsets of set A

• $\mathrm { K } = \{ \mathrm { b } , \mathrm { c } , \mathrm { d } \}$
• $\mathrm { L } = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } \}$
• $\mathrm { M } = \{ \mathrm { c } , \mathrm { d } , \mathrm { e } \}$

are closed under the $\Delta$ operation?
A) Only K
B) Only L
C) K and L
D) K and M
E) L and M