Problems

1. Using only two-input NOR gates, show how AND, OR and NAND gates can be made.
2. The binary addition of two 2-bit numbers (with carry bits) looks like the following:

   

   1. Write a truth table expressing the outputs and as a function of , and .
   2. Write an algebraic statement in Boolean algebra describing this truth table.
   3. Implement this statement using standard (AND, OR, EX-OR and inverter gates).
3. If the 3-bit binary number A B C represents the digits 0 to 7:
   1. Make a truth table for A, B, C and Q, where Q is true only when an odd number of bits are true in the number.
   2. Write a statement in Boolean algebra for Q.
   3. Convert this equation to one that can be mechanized using only two XOR gates. Draw the resulting circuit.
4. You need to provide a logic signal to control an experiment. The experiment is controlled by the four signals A, B, C and D, which make up the data word A B C D. The control line Q should be set high only if this data word takes on the values 1, 3, 5, 7, 11 or 13.
   1. Write a truth table for this function.
   2. Using boolean algebra, write an expression indicating when Q is true. Your statement should include one logical statement for each of the six true conditions, each separated by the OR function. Therefore this statement should utilize a number of AND/NAND functions, and five OR statements.
   3. Simplify this statement so that it can be implemented using two two-input AND gates, one OR gate, one exclusive OR gate and one logical inverter. The expression has the form , where stands for any of the four input signals or their logical inverses.
   4. Implement this simplified function using logic gates.