Consider the four-variable Boolean function, f(w, x, y, z) = Σm(0, 2, 5, 7, 8, 10, 13, 14, 15) with w as MSB and z as LSB. Which of the following expressions is/are the valid form(s) of f(w, x, y, z)?

The correct options are:
1. $\overline{x}\overline{z}+xz+wxy$
2. $\overline{x}\overline{z}+xz+wy\overline{z}$

To determine which expressions are valid forms of the Boolean function $f(w,x,y,z)=\Sigma m(0,2,5,7,8,10,13,14,15)$, we can analyze the minterms using a Karnaugh map (K-map).

Let us represent the 4-variable K-map with $wx$ along the rows and $yz$ along the columns:
- Rows: $\overline{w}\overline{x}$ (00), $\overline{w}x$ (01), $wx$ (11), $w\overline{x}$ (10)
- Columns: $\overline{y}\overline{z}$ (00), $\overline{y}z$ (01), $yz$ (11), $y\overline{z}$ (10)

The minterm numbers correspond to the K-map cells as follows:
- Row 00: m0 (0000), m1 (0001), m3 (0011), m2 (0010)
- Row 01: m4 (0100), m5 (0101), m7 (0111), m6 (0110)
- Row 11: m12 (1100), m13 (1101), m15 (1111), m14 (1110)
- Row 10: m8 (1000), m9 (1001), m11 (1011), m10 (1010)

Placing 1s in the cells corresponding to the given minterms: {0, 2, 5, 7, 8, 10, 13, 14, 15}, we get:
- Row 00 ($\overline{w}\overline{x}$): 1 at col 00 (m0), 1 at col 10 (m2)
- Row 01 ($\overline{w}x$): 1 at col 01 (m5), 1 at col 11 (m7)
- Row 11 ($wx$): 1 at col 01 (m13), 1 at col 11 (m15), 1 at col 10 (m14)
- Row 10 ($w\overline{x}$): 1 at col 00 (m8), 1 at col 10 (m10)

Let us group the cells to find the prime implicants:
1. Grouping the four corners: m0, m2, m8, m10. This quad gives the term:
$\overline{x}\overline{z}$
2. Grouping the quad in the columns with $z=1$: m5, m7, m13, m15. This quad gives the term:
$xz$

After these two groups, the only remaining uncovered minterm is m14 (1110).
We can cover m14 using one of two possible groups:
- Grouping m14 with m15 (1111): The group is {m14, m15}, which corresponds to row 11 ($wx$) and columns 11 and 10 ($y$). This yields the term:
$wxy$
- Grouping m14 with m10 (1010): The group is {m10, m14}, which corresponds to rows 11 and 10 ($w$) and column 10 ($y\overline{z}$). This yields the term:
$wy\overline{z}$

Combining these groups gives two valid minimal sum-of-products (SOP) expressions for $f$:
1. $f=\overline{x}\overline{z}+xz+wxy$
2. $f=\overline{x}\overline{z}+xz+wy\overline{z}$

Both of these expressions are minimal SOP forms and represent valid forms of $f(w,x,y,z)$.