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a) Consider the full adder function:
sum(c i-1, xi , yi ) = (ci , si )
Where xi , yi are the ith bits of the binary numbers X and Y, ci-1 is the carry in and the outputs are: ci , the carry out and si is the sum output.
Write the equations of both ci and si in conjunctive normal form (CNF),
i.e. in standard product of sums (SPOS).
b) As a result of a) above, design the corresponding circuit for the full adder, using only the NOR gates..
Suppose that, in the circuit for a 2-bit comparator, we wanted to allow for inputs from the
higher byte comparisons, xH < yH, xH = yH, xH > yH, so that we could now use a 2-bit
comparator in cascades for building comparators for numbers with more than 2 bits.
a) Re-write the equations for a 2-bit comparator that would allow for the inputs: xH < yH,
xH = yH, xH > yH.
b) Re-draw the corresponding circuit diagram.
Question 2:
Consider the following narration:
If equality of opportunity is to be achieved then those people previously disadvantaged should be
given special opportunities. If those people previously disadvantaged should be given special
opportunities then some people should receive preferential treatment. If some people receive
preferential treatment then equality of opportunity is not to be achieved. Some people are not going
to receive preferential treatment. Therefore, equality of opportunity is not to be achieved.
a) Encode the argument in Propositional Calculus.
b) As a result of a) above, use logical rules of inference to show that the given argument is sound/valid.
HINT: For any statement forms, P and Q P \rightarrow Q = ¬Q \rightarrrow ¬P.Draw hasse diagram representing the positive divisor of 36 and show digraph to It?
Let A = {a,b,c,d}, B = {1,2,3}, and R = {(a,2), (b, 1), (c, 2), (d, 1)}.
(a)Is R a function?
(b)Is R−1 a function?
Explain your answers.
Let A = {1,2,3,4,5,6} and let p1 = (3,6,2) and p2 = (5, 1, 4) be permutations of A.
(a) Compute p1 ◦ p2 and write the result as a product of cycles and as the product of
transpositions.
(b) Compute p−1 ◦ p−1
The market for lemon has 10 potential consumers, each having an individual demand curve
P = 101 - 10Qi
, where P is price in dollars per cup and Qi is the number of cups demanded
per week by the i
th consumer. Find the market demand curve using algebra. Draw an
individual demand curve and the market demand curve. What is the quantity demanded by
each consumer and in the market as a whole when lemon is priced at P = $1/cup?
Let A = {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 15} and define relations R and S from X to Y as
follows:
for all (x,y) ∈A × B, x R y ⇔ y % x
for all (x,y) ∈A × B, x S y ⇔ (y – 4)+2 = x
State explicitly which ordered pairs are in A × B, R, S and type of relation
Let A={0, 1,2}×{2, 5,8}={(0, 2), (0, 5), (0, 8), (1, 2), (1, 5), (1, 8), (2, 2), (2, 5), (2, 8) }.
A partial order relation R on A is defined by (a, b) R(c, d) if and only if (a+b) divides (c+d). Draw a hasse diagram for poset A.
State the converse and contrapositive of each of the following implications.
(a) If it does not rain tonight, I will go fishing tomorrow.
(b) If it rains tonight, then I will stay at home.
#331389 on Nov 2022