Question

CD D
15. Are the two expressions below equivalent?
Z = ( ๐‘ฟ ฬ…ฬ…ฬ…ฬ…ฬ…ฬ…
ฬ…
ยท ๐’€
ฬ…
)
Z = ( X + Y )
16. Write the logic expression for the following Karnaugh
Map in its simplest SOP form.
AB
Z
00 01 11 10 00 1 0 1 0 01 0
1
0
1 11 1 0 1 0 10
0
1
0
1

Answers

GPT-4o mini
To address your two inquiries:

**15. Are the two expressions below equivalent?**

Given:
1. \( Z = (\overline{X} \cdot \overline{Y}) \)
2. \( Z = (X + Y) \)

To determine if these two expressions are equivalent, we can analyze each expression using truth values.

1. For \( Z = (\overline{X} \cdot \overline{Y}) \):
- This means \( Z = 1 \) only when both \( X \) and \( Y \) are 0.
- Truth table:
| X | Y | \( \overline{X} \) | \( \overline{Y} \) | \( \overline{X} \cdot \overline{Y} \) |
|---|---|--------------|--------------|--------------------------|
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 | 0 |

2. For \( Z = (X + Y) \):
- This means \( Z = 1 \) if either \( X = 1 \) or \( Y = 1 \) (at least one input is true).
- Truth table:
| X | Y | \( X + Y \) |
|---|---|----------|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |

From the truth tables above:

- For \( (X, Y) = (0, 0) \), both expressions evaluate to 1.
- For \( (0, 1) \) or \( (1, 0) \) or \( (1, 1) \), the expressions evaluate to 0 and 1 respectively.

**Conclusion**: The expressions are **not equivalent**. The first expression outputs true only when both inputs are false, while the second expression outputs true whenever at least one input is true.

---

**16. Write the logic expression for the Karnaugh Map in its simplest SOP form.**

From the provided Karnaugh Map layout, placing "1"s in the appropriate cells based on the values indicates the inputs that contribute to the SOP (Sum of Products) expression.

The Karnaugh Map appears to have "1"s in the following positions:
- Cell (00, 00) (Top row, first column)
- Cell (00, 11) (Top row, third column)
- Cell (01, 10) (Middle row, second column)
- Cell (11, 01) (Bottom row, second column)
- Cell (11, 11) (Bottom row, third column)

1. The positions for the cells correspond to the following minterms:
- \( A'B' \) (for 00)
- \( AB \) (for 11)
- \( A'B \) (for 01)
- \( AB' \) (for 10)

Thus, the SOP expression will be:

**SOP Expression:**
\[ Z = A'B' + A'B + AB' + AB \]

Now let's simplify the expression:

Combining the terms:
\[ Z = A'B' + A'B + AB' + AB = A'B' + A'(B + B') + AB = A'B' + A' + AB \]
\[ = A' + AB \]

Thus, the simplest SOP form becomes:

\[ Z = A' + AB \]

This is the final expression for the Karnaugh Map you described.

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