سوال ۷۷ کنکور ارشد مهندسی کامپیوتر ۱۴۰۵

Evaluate the function

Given $F = x\cdot(y\oplus z)' + y z'$. Note that $(y\oplus z)'$ is the XNOR (true when $y=z$). Over all eight input combinations, $F=1$ exactly for the minterms $m_2, m_4, m_6, m_7$.

Minimize the POS (the right form for NOR gates)

A two-level NOR–NOR network natively realizes a product-of-sums, so we group the zeros of $F$ (minterms $m_0,m_1,m_3,m_5$):

$F' = x'y' + x'z + y'z$

Complementing with De Morgan gives the minimal POS:

$F = (x+y)(x+z')(y+z')$

Map to NOR–NOR

$F = \overline{\overline{(x+y)}+\overline{(x+z')}+\overline{(y+z')}} = \mathrm{NOR}\big(\mathrm{NOR}(x,y),\ \mathrm{NOR}(x,z'),\ \mathrm{NOR}(y,z')\big)$

• Level 1: three 2-input NOR gates producing $(x+y)'$, $(x+z')'$, $(y+z')'$.
• Level 2: one 3-input NOR gate collecting the three.

Each variable and its complement (here $z'$) are given, so no extra inverters are required. Total: three 2-input NOR gates + one 3-input NOR gate — option 3.

Exam trick: For a NOR-only realization, minimize the POS (group the 0s). The number of sum terms equals the number of first-level 2-input NORs, and one final NOR with that many inputs collects them.