In a digital combinational circuit with $$4$$ inputs $$(A, B, C, D),$$ it is required to obtain an output of logical $$1$$ only for the input combination

$$(A = 1; B = C = D = 0).$$ It is known that the following combinations of input are forbidden:

$$ABCD = 1010, 1011, 1100, 1101, 1110, 1111$$
Evaluate the logical expression for the output and realize the same with two input $$NAND$$ gates. Assume that complements of inputs are not available.

A $$3$$-input $$2$$-output priority encoder has the following truth table where $$'X'$$ indicate don't care conditions. Realize the logic using $$NAND$$ gates and inverters.