Why is the result of (a+b+c')(a'b'+c) not 1? [closed] - boolean-logic

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So I got his problem as homework in my course, when I solved it the result I came up with was 1, but everywhere I check the solution stops at line 4 is if it is the final solution, but I can't spot the error in my logic for some reason!
Line1: (a+b+c')(a'b'+c)
Line2: =aa'b'+ba'b'+c'a'b'+ac+bc+c'c
Line3: =0+0+c'a'b'+ac+bc+0
Line4: =c'a'b'+ac+bc
Line5: =c'a'b'+c(a+b)
Line6: =c'+c(a'b'+(a+b))
Line7: =1*(a'b'+(a+b))
Line8: =1

Answer 1

You can do a very little bit better if XOR (^) is a primitive operation in your system:
Line1: (a+b+c')(a'b'+c)
Line2: = aa'b'+ba'b'+c'a'b'+ac+bc+c'c
Line3: = 0+0+c'a'b'+ac+bc+0
Line4: = c'a'b'+ac+bc
Line5: = c'a'b'+c(a+b)
Line6: = c'a'b'+c(a'b')'
Line7: = c^(a'b')
Your error is the following:
Line5: = c'a'b'+c(a+b)
Line6: = c'+c(a'b'+(a+b))
Clearly, lines 5 and 6 do not show equivalent expressions because c=0 satisfies the second regardless of a and b whereas c=0, a=1 does not satisfy the first.