ICS 6B, Homework 9 wk9(1) 2. Find the values, if any, of the Boolean variable that satisfy these equations. x a) 1 = 0 b) + = 0 x x x ・ c) 1 = d) = 1 x x x ・ ・ 4. a) Show that + = 1. ( ) (1 ) ・ ・ Translate the equation in part (a) into a propositional equivalence by changing each 0 into an b) F, each 1 into a each Boolean sum into a disjunction, each Boolean product into a conjunction, T, each complementation into a negation, and the equals sign into a propositional equivalence sign. Use a table to express the values of each of these Boolean functions. 6. a) F(x, y, z) = F(x, y, z) = xz + c) 20. Verify the first distributive law: + = + + x yz (x y)(x z) wk9(2) 4. Find the sum-of-products expansions of the Boolean function that equals 1 if and only if F(x, y, z) a) = 0. b) = 0. x xy c) + = 0. d) = 0. x y xyz 12. Express each of these Boolean functions using the operators and ・ − a) + + x y z c) wk9(3) 4. Find the output of the given circuit. 6. Construct circuits from inverters, AND gates, and OR gates to produce these outputs. a) + y b) x c) xyz + d) 10. Construct a circuit for a half subtractor using AND gates, OR gates, and inverters. A half subtractor has two bits as input and produces as output a difference bit and a borrow.
ICS 6B, Homework 9 wk9(1) 2. Find the values, if any, of the Boolean variable x that satisfy these equations. a) x ・ 1 = 0 b) x + x = 0 c) x ・ 1 = x d) x ・ = 1 4. a) Show that ( ・ ) + (1 ・ ) = 1. b) Translate the equation in part (a) into a propositional equivalence by changing each 0 into an F, each 1 into a T, each Boolean sum into a disjunction, each Boolean product into a conjunction, each complementation into a negation, and the equals sign into a propositional equivalence sign. 6. Use a table to express the values of each of these Boolean functions. a) F(x, y, z) = c) F(x, y, z) = xz + 20. Verify the first distributive law: x + yz = (x + y)(x + z) wk9(2) 4. Find the sum-of-products expansions of the Boolean function F(x, y, z) that equals 1 if and only if a) x = 0. b) xy = 0. c) x + y = 0. d) xyz = 0. 12. Express each of these Boolean functions using the operators ・ and − a) x + y + z c) wk9(3) 4. Find the output of the given circuit. 6. Construct circuits from inverters, AND gates, and OR gates to produce these outputs. a) + y b) x c) xyz + d) 10. Construct a circuit for a half subtractor using AND gates, OR gates, and inverters. A half subtractor has two bits as input and produces as output a difference bit and a borrow.
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