# Download e-book for kindle: Algebra and number theory. An integrated approach by Dixon M., Kurdachenko L., Subbotin I.

By Dixon M., Kurdachenko L., Subbotin I.

ISBN-10: 0470496363

ISBN-13: 9780470496367

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Additional info for Algebra and number theory. An integrated approach

Example text

The switching function of the opposite circuit to that defining f is f , where f (A1 , . . , An ) = (f (A1 , . . , An )) . 19. The set of n-variable switching functions forms a boolean algen bra (Fn , ∧, ∨, ) that contains 22 elements. Proof. It can be verified that (Fn , ∧, ∨, ) satisfies all the axioms of a boolean algebra. The zero element is the function whose image is always 0, and the unit element is the function whose image is always 1. 11. For example, f6 (A, B) = 0 if A = B, and 1 if A = B.

Write down the tables for the NOR and NAND operations on the set P ({a, b, c}). 65. Can every switching circuit be built out of AND and NOT gates? 66. (a) Design a half adder using five NOR gates. (b) Design a half adder using five NAND gates. 67. 38. 68. Design a NOR circuit that will produce a parity check symbol for four binary input digits; that is, the circuit must produce a 0 if the inputs contain an even number of 1’s, and it must produce 1 otherwise. 69. One of the basic types of components of a digital computer is a flip-flop.

S2 s1 s0 . 1111 101 1111 ← carry digits → 15 5 1 . . a2 a1 a0 . . b2 b1 b0 . . c2 c1 10100 20 . . s2 s1 s0 binary addition decimal addition Let us first design a circuit to add a0 and b0 to obtain s0 and the carry digit c1 . This is called a half adder. 16. For example, in binary arithmetic, 1 + 1 = 10, which means that if a0 = 1 and b0 = 1, s0 = 0, and we have to carry c1 = 1. 16 that c1 = a0 ∧ b0 and s0 = (a0 ∧ b0 ) ∨ (a0 ∧ b0 ). 26. 16. 26. Circuits for the half adder. 17. 27. a ′i ai Venn diagrams for a full adder.