There are four Boolean variables x1, x2, x3 and x4. The

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There are four Boolean variables x₁, x₂, x₃ and x₄. The following functions are defined on sets of them
f (x₃, x₂, x₁) = Σ m (3, 4, 5)
g (x₄, x₃, x₂) = Σ m (1, 6, 7)
h (x₄, x₃, x₂, x₁) = fg
Then h (x₄, x₃, x₂, x₁) is—

1. Σ m (3, 12, 13)
2. Σ m (3, 6)
3. Σ m (3, 12)
4. 0

Correct Option: A

Given
f = (X₃, X₂, X₁) = Σm (3, 4, 5)
or
f = X₃ X₂ X₁ + X₃X₂X₁
= X₃X₂X₁ + X₃X₂
and g(X₄, X₃, X₂) = Σm(1,6,7)
or
g = X₄X₃X₂ + X₄X₃X₂ + X₄ X₃ X₂
= X₄X₃X₂X₁ + X₄X₃
Now,
f_g = (X₃ X₂ X₁ + X₃X₂) (X₄X₃ X₂ + X₄ X₃)
= X₄X₃X₂X₁ + X₄X₃X₂
so,h(X₄, X₃, X₂,X₁)= f_g
or,h(X₄, X₃, X₂,X₁)=X₄X₃X₂X₁ + X₄X₃X₂
or,h(X₄, X₃, X₂,X₁)= X₄X₃X₂X₁
+ X₄X₃X₂(X₁ + X₁)
or,h(X₄, X₃, X₂,X₁)= X₄X₃X₂X₁
+ X₄X₃X₂X₁ + X₄X₃X₂X₁
or,h(X₄, X₃, X₂,X₁)= Σm(3,13,12)
Hence alternative (A) is the correct choice.