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$f(a,b,c)=\sum m(1,2,4,7)$ is a majority circuit and it is XOR

$f(a,b,c)=\sum m(1,2,4,7) f={a}'{b}'c+{a}'b{c}'+a{b}'{c}'+abc$

$f(a,b,c)={a}'{b}'c+{a}'b{c}'+a{b}'{c}'+abc$

after solving the equation, we get

$f(a,b,c)=a\oplus b\oplus c$
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Either we can take each minterm from the k-map and write sop, simplify to obtain exclusive OR as there are no groups  formed or we can see the relationship Between the Boolean  variables resulting in a 1 in the k-map and we can deduce from the option .
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In any k-map, if all minterms (1) are diagonal to each other, then there is only one minterm  in each valid sub-cube and thus the function representing that k-map is definitely "Ex-Nor" or "Ex-Or"

Now the question is that when it will ex-or and when it will be ex-nor?

when the first 1 is present in the 0th cell i.e upper left cell, then it will represent Ex-Nor. As shown below.

                     1                     0            1                0
                     0                    1            0                1

 

F = A $\bigodot$ B $\bigodot$ C


when the first 1 is present in the 1st cell, then it will represent Ex-or. As shown below.

                    0                       1                         0                 1
                    1                        0                         1                 0

F = A $\bigoplus$ B $\bigoplus$ C 

 

SIMILAR FOR FOUR VARIABLE CASE.

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