Answer:

The Karnaugh map for a 3-variable function is as follows:

We can group the '1's as follows:

• The two '1's in the first row (000 and 001) can be grouped.
• The two '1's in the fourth column (011 and 111) can be grouped.
• The '1' in the third row, second column (101) can be grouped with the '1' in the third row, third column (110).

This gives us the simplified expression: f(x, y, z) = (A'C' + A'BC) + (AB'C + ABC) = A'C' + A'BC + AB'C + ABC = A'C' + AB'C + BC

Or, using Boolean algebra: f(x, y, z) = x'z' + x'yz + xy'z + xyz

JK Flip-Flop for a Counter: A JK flip-flop can be used to create a counter by cascading multiple JK flip-flops. The J and K inputs are connected to generate the desired state transitions. For example, to create a 2-bit counter, the J and K inputs of the flip-flops are connected to generate the sequence 00, 01, 10, 11, and then back to 00. The clock signal triggers the state transitions.

• Operation: The JK flip-flop's ability to toggle its state when both inputs are '1' is crucial for counter functionality. The specific combination of J and K inputs determines the sequence of states.
• Design: Requires careful wiring to ensure the correct sequence of state transitions. The design can become complex for larger counters.

D Flip-Flop for a Counter: A D flip-flop is a simpler alternative for implementing a counter. The D input is connected to the desired next state of the counter. The clock signal triggers the transfer of the value on the D input to the Q output. To create a counter, the D input is set to the desired next state for each clock pulse.

• Operation: The D flip-flop directly captures the value on the D input at the active clock edge. This makes it easier to implement counters with predictable state transitions.
• Design: Generally simpler to design than a counter based on JK flip-flops, especially for larger counters. The D flip-flop's behavior is more straightforward to analyze.

Comparison:

SR Flip-Flop: The SR flip-flop is a basic sequential logic circuit with two inputs: Set (S) and Reset (R).

• Operation: When S is asserted (typically high), the flip-flop transitions to a '1' state. When R is asserted (typically high), it transitions to a '0' state. If both S and R are asserted simultaneously, the output state is undefined (often considered a metastable state).
• Truth Table:

  S	R	Q(t+1)
  0	0	Q(t)
  0	1	0
  1	0	1
  1	1	Undefined

• Advantages: Simple design, easy to understand.
• Disadvantages: The 'S' and 'R' inputs can conflict, leading to an undefined state. This makes it unsuitable for applications requiring reliable state transitions.
• Suitability: Limited use in practical digital circuits due to the undefined state. May be used in specific applications where the undefined state is acceptable or can be managed.

JK Flip-Flop: The JK flip-flop addresses the problem of the SR flip-flop by providing a third input, J (Justify), which eliminates the undefined state.

• Operation:

  • When J=0 and K=0, the flip-flop holds its current state.
  • When J=0 and K=1, the flip-flop resets to '0'.
  • When J=1 and K=0, the flip-flop sets to '1'.
  • When J=1 and K=1, the flip-flop toggles its state (flips from '0' to '1' or '1' to '0').

• Truth Table:

  J	K	Q(t+1)
  0	0	Q(t)
  0	1	0
  1	0	1
  1	1	!Q(t)

• Advantages: Eliminates the undefined state of the SR flip-flop. Provides a clear and predictable state transition.
• Disadvantages: Slightly more complex than the SR flip-flop.
• Suitability: Widely used in digital circuits due to its reliability and predictable behavior. Suitable for a broad range of applications, including registers, counters, and memory elements.