Find out the canonical equivalent SOP for the function Y = AC + AB + BC.*Σm (3, 5, 6, 7)Σm (3, 6, 7)Σm (4, 5, 6)Σm (5, 6, 7)

1. Break Down the Problem

We need to find the canonical equivalent Sum of Products (SOP) for the given function $Y = AC + AB + BC$ and the minterms provided. The minterms are represented as $\Sigma m(n)$, which denotes a sum of the minterms for which the function outputs true (1).

2. Relevant Concepts

To express the function in canonical SOP form:

• Identify the variables involved (A, B, C).
• Convert the minterm indices into binary to determine the corresponding product terms.
• Combine the terms to form the canonical SOP.

Given minterms:

• $\Sigma m (3, 5, 6, 7)$
• $\Sigma m (3, 6, 7)$
• $\Sigma m (4, 5, 6)$
• $\Sigma m (5, 6, 7)$

3. Analysis and Detail

• Identify the binary representation of each minterm:

  \begin{align*} \text{Minterm 3} & : 011 \rightarrow A'B C \\ \text{Minterm 5} & : 101 \rightarrow AB C' \\ \text{Minterm 6} & : 110 \rightarrow A B C \\ \text{Minterm 7} & : 111 \rightarrow A B C \\ \text{Minterm 4} & : 100 \rightarrow A B' C \\ \end{align*}

• Now we can compile the product terms from the relevant minterms:

  • From $\Sigma m (3, 5, 6, 7)$: $A'BC, ABC', AB C, A B C$
  • From $\Sigma m (3, 6, 7)$: $A'BC, AB C, A B C$
  • From $\Sigma m (4, 5, 6)$: $AB'C, ABC', AB C$
  • From $\Sigma m (5, 6, 7)$: $AB C', AB C, A B C$

• Combining unique product terms leads us to: $\text{Canonical SOP} = A'BC + AB'C + ABC' + ABC$

4. Verify and Summarize

In canonical form, each unique term represents the combinations that give a '1' for the function. We have considered all specified minterms, and thus, the function is completely represented.

Final Answer

The canonical equivalent SOP for the function $Y$ is:

$Y = A'BC + AB'C + ABC' + ABC$