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Answer the following using a truth table and a Karnaugh map.
A Football Association coach analyses the criteria for a win/draw of his team depending on the…
A Football Association coach analyses the criteria for a win/draw of his team depending on the following conditions.
• If the Centre and Forward players perform well but Defenders do not perform well.
OR
• If Goalkeeper and Defenders perform well but the Centre players do not perform well.
OR
• If all perform well.
The inputs are:
(In all the above cases, 1 indicates yes and 0 indicates no.)
Output: X - Denotes the win/draw criteria [1 indicates win/draw and 0 indicates defeat in all cases]
| INPUTS | |
|---|---|
| C | Centre players perform well |
| D | Defenders perform well |
| F | Forward players perform well |
| G | Goalkeeper perform well |
(i)[5.0]
Draw the truth table for the inputs and outputs given above and write the SOP expression for $X(C, D, F, G)$.
(ii)(a)[2.5]
Reduce the above expression $X (C, D, F, G)$ by using 4-variable Karnaugh map, showing the various groups (i.e. octal, quads and pairs).
(ii)(b)[2.5]
Draw the logic gate diagram for the reduced expression. Assume that the variables and their complements are available as inputs.
Draw: logic gate diagram for the reduced expression
Answer
Answer
AI3(i): Truth table for $X(C,D,F,G)$:
From the three given conditions:
Condition 1 (Centre, Forward well; Defenders not well): $C \cdot D' \cdot F$
Condition 2 (Goalkeeper, Defenders well; Centre not well): $C' \cdot D \cdot G$
Condition 3 (all perform well): $C \cdot D \cdot F \cdot G$
SOP expression: $X(C,D,F,G) = C \cdot D' \cdot F + C' \cdot D \cdot G + C \cdot D \cdot F \cdot G$
3(ii): Plotting the minterms of $X = CD'F + C'DG + CDFG$, i.e. $\Sigma(5,7,10,11,15)$, on a 4-variable K-map (rows CD: 00,01,11,10; columns FG: 00,01,11,10) gives three pairs (no quad or octet is possible for this function):
Pair (10,11): $C=1,D=0,F=1$, G varies $\Rightarrow CD'F$
Pair (5,7): $C=0,D=1,G=1$, F varies $\Rightarrow C'DG$
Pair (7,15): $D=1,F=1,G=1$, C varies $\Rightarrow DFG$
Reduced (minimal) SOP: $X(C,D,F,G) = CD'F + C'DG + DFG$
Logic gate diagram: three 3-input AND gates feeding one 3-input OR gate.
| C | D | F | G | X |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 |
From ISC 2024 Specimen Computer Science Paper 1, question 3.
Check your working with the Boolean algebra solver: the steps law by law, the K-map and the logic circuit for any expression.