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To express the given function F(A, B, C, D) as a sum of minterms, we need to consider the terms where the output of the function is equal to 1.

F(A, B, C, D) = B'D + A'D + BD

We can express each of the terms as a minterm:

B'D = A'B'CD' A'D = A'BCD' BD = ABCD + A'BC'D + AB'CD'

So the function F(A, B, C, D) can be expressed as the sum of these minterms:

F(A, B, C, D) = A'B'CD' + A'BCD' + ABCD + A'BC'D + AB'CD'
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F(A, B, C, D) = B'D + A'D + BD = (B’ + A’ + B)D = (1 + A’)D = D   [B’+B = 1  ;   1+A’ = 1]

=>  F(A,B,C,D) = D = $\sum (1,3,5,7,9,11,13,15)$

Now let’s see how D = $\sum (1,3,5,7,9,11,13,15)$

A B C D minterms
0 0 0 1 (0001)$_{2}$ = (1)$_{10}$ (A’B’C’D)
0 0 1 1 (0011)$_{2}$ = (3)$_{10}$ (A’B’CD)
0 1 0 1 (0101)$_{2}$ = (5)$_{10}$ (A’BC’D)
0 1 1 1 (0111)$_{2}$ = (7)$_{10}$ (A’BCD)
1 0 0 1 (1001)$_{2}$ = (9)$_{10}$ (AB’C’D)
1 0 1 1 (1011)$_{2}$ = (11)$_{10}$ (AB’CD)
1 1 0 1 (1101)$_{2}$ = (13)$_{10}$ (ABC’D)
1 1 1 1 (1111)$_{2}$ = (15)$_{10}$ (ABCD)

(We consider A’ as 0 and A as 1)

So, D = $\sum (1,3,5,7,9,11,13,15)$

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