05s_cpe422_hw1_solution - The University of Alabama in...

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The University of Alabama in Huntsville Electrical & Computer Engineering CPE/EE 422/522 01 Homework #1 Solution Spring 2005 1. (10 points) Prove that the following identities are valid using algebraic manipulation. a. ab’ + bc’ + ca’ = a’b + b’c + c’a ab’ + bc’ + ca’ = ab’(c+c’)+bc’(a+a’)+ca’(b+b’) = ab’c+ab’c’+abc’+a’bc’+a’bc+a’b’c = a’b(c’+c)+b’c(a+a’)+ac’(b’+b) a’b+b’c+c’a b. ab + a’c + bcd = ab + a’c ab + a’c + bcd = ab(c + c’)(d + d’) + a’c(b’ + b)(d + d’) + bcd(a’ + a) = abc’d’+abc’d+abcd’+abcd+a’b’cd’+a’b’cd+a’bcd’+a’bcd+a’bcd+abcd = abc(d’+d)+abc(d’+d)+a’b’c(d’+d) + a’bc(d’+d) = ab(c’+c)+a’c(b’+b) = ab + a’c 2. (10 points) Find the complements of the following functions. a. f = a + bc f’ = (a + bc)’ = a’(bc)’ = a’(b’ + c’) = a’b’ + a’c’ b. f = ab + b’c + ca’d f’ = (ab + b’c + ca’d)’ = (ab)’(b’c)’(ca’d)’ = (a’ + b’)(b + c’)(c’ + a + d’) = (a’b + a’c’ +
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05s_cpe422_hw1_solution - The University of Alabama in...

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