Final answer:
The allowed spectroscopic notations are 1s¹, 4s², and 3p²; the other notations violate quantum number rules due to incorrect subshells or exceeded electron capacity.
Step-by-step explanation:
The allowed spectroscopic notations respecting the rules of quantum numbers are 1s¹, 4s², and 3p². Notations such as 1d³ (1ď³) and 6h²Σ0 are not possible because the d subshell doesn't exist for n=1, and there are no h subshells in known chemistry or physics. The principal quantum number, n, specifies the shells (n=1,2,3...) while the azimuthal quantum number, l, defines the subshells associated with each shell (s, p, d, f...).
The notation (c) 4s² denotes the s subshell of the 4th energy level with 2 electrons, adhering to the principle that each orbital can accommodate up to 2 electrons. Notation (a) 1s¹ correctly represents one electron in the s subshell of the first energy level. However, notation (e) 6h²Σ0 violates the rule that h subshells do not exist and notation (d) 3p· incorrectly represents 7 electrons in the p subshell which can only hold a maximum of 6 electrons, violating the rule of a maximum of 2(2l+1) electrons per subshell.The set notation that denotes the empty set is represented as {} or ⌀. In this case, the set notation (a) { z : z is a horse and z has 6 legs } does not denote the empty set because there are no horses with 6 legs. However, the set notations (b) a ∈ r : a² + 2a + 2 = 0 and (c) n ∈ ℕ : n² + n + 11 is not prime do denote the empty set. This is because the equations in both sets have no solutions.