Let f : R \(\to\) R be the Signum Function defined as \(f(x) = \begin{cases} 1, & \quad \text x>0 \\ 0, & \quad x=0 \\ -1, &\quad x<0 \end{cases}\)
and \(g: R \to R\) be the Greatest Integer Function given by g (x)= [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0,1]?
It is given that,f : R\(\to\)R is defined as \(f(x) = \begin{cases} 1, & \quad \text x>0 \\ 0, & \quad x=0 \\ -1, &\quad x<0 \end{cases}\)
Also, g : R \(\to\) R is defined as g (x) = [x], where [x] is the greatest integer less than or equal to x.
Now, let x ∈ (0, 1].
Then, we have:
[x] = 1 if x = 1 and [x] = 0 if 0 < x < 1.
\(\therefore\) \(f(x) = f(g(x))=f([x]) \begin{cases} f(1), & \quad \text{if } x=1 \\ f(0), & \quad \text{if } x\in (0,1) \end{cases}\)\(= \begin{cases} 1, & \quad \text{if } x=1 \\ 0, & \quad \text{if } x\in (0,1) \end{cases}\)
gof (x) = g (f(x))= g (1) [x>0]
=[1]=1.
Thus, when x ∈ (0, 1), we have fog (x) = 0 and gof (x) = 1.
Hence, fog and gof do not coincide in (0, 1].
A school is organizing a debate competition with participants as speakers and judges. $ S = \{S_1, S_2, S_3, S_4\} $ where $ S = \{S_1, S_2, S_3, S_4\} $ represents the set of speakers. The judges are represented by the set: $ J = \{J_1, J_2, J_3\} $ where $ J = \{J_1, J_2, J_3\} $ represents the set of judges. Each speaker can be assigned only one judge. Let $ R $ be a relation from set $ S $ to $ J $ defined as: $ R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y, x \in S, y \in J\} $.
The correct IUPAC name of \([ \text{Pt}(\text{NH}_3)_2\text{Cl}_2 ]^{2+} \) is: