Correct : c,d

Explanation:
Let us evaluate the output of each logic gate step-by-step using minterm sets to find the final expression for Y.
1. Analyze the AND Gate Output:
The inputs to the AND gate are f₁ and f₂. The logical AND operation corresponds to the intersection (∩) of their minterm sets (the minterms present in both functions).
• f₁ = {0, 2, 3, 5, 7, 8, 11, 13}
• f₂ = {1, 3, 5, 7, 11, 13, 15}

    Output_AND = f₁ ∩ f₂ = {3, 5, 7, 11, 13}

2. Analyze the OR Gate Output:
The inputs to the OR gate are f₃ and f₄. The logical OR operation corresponds to the union (∪) of their minterm sets (combining all elements from both functions).
• f₃ = {0, 1, 4, 11}
• f₄ = {0, 2, 6, 13}

    Output_OR = f₃ ∪ f₄ = {0, 1, 2, 4, 6, 11, 13}

3. Analyze the Final XOR Gate Output (Y):
The final output Y is the XOR of Output_AND and Output_OR. The logical XOR operation corresponds to the symmetric difference (Δ) of the two sets (elements that are in either of the sets, but not both).
• Output_AND = {3, 5, 7, 11, 13}
• Output_OR = {0, 1, 2, 4, 6, 11, 13}

Let's find the common elements (intersection) first:
• Common minterms shared by both paths = {11, 13}

Filter out these common elements from the combined set to get the symmetric difference:
    Y = ∑(0, 1, 2, 3, 4, 5, 6, 7)

4. Evaluate the Minterm Options:
• Option (a): Y = ∑(0, 1, 2, 11, 13) — Incorrect.
• Option (c): Y = ∑(0, 1, 2, 3, 4, 5, 6, 7) — Correct.

5. Convert to Maxterm Form:
Since this is a 4-variable function, the total possible minterms span from 0 to 15. The product-of-maxterms (∏) expression consists of all the numbers missing from our sum-of-minterms list:
    Missing terms = {8, 9, 10, 11, 12, 13, 14, 15}
    Y = ∏(8, 9, 10, 11, 12, 13, 14, 15)

• Option (b): Y = ∏(3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15) — Incorrect.
• Option (d): Y = ∏(8, 9, 10, 11, 12, 13, 14, 15) — Correct.

6. Conclusion:
Options c and d are the logically valid statements for this digital network.