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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.
Prove or disprove the validity of the argument"every living things is a plant or an animal",Kamal's dog is alive and it is not a plant",All animals have heart",Hence "Kamal's dog has a heart".
a) Consider the full adder function: i-1 i i i i sum(c , x , y ) = (c , s) Where xi i i-1 , y are the i bits of the binary numbers X and Y, c is the carry in and the outputs are: th ci i , the carry out and s is the sum output. Write the equations o i i f both c and s 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..
a) Consider the whole of the English words set. Suppose an English word x is related to another English word y if x and y begin with the same letter. i) Show that this is an equivalence relation. ii) Compute C(quadratic) and C(rhombus) iii) How many equivalence classes are there in all, and why? iv) What is the partition of the English words under this relation?
b) Consider Z, the set of integers. Suppose we define the relation: x is related to y if x - y > 3, x, y 0 Z. Determine whether or not the relation is i) reflexive ii) symmetric
a) Consider the following functional relation, f, defined as:
f : R \rightarrow R, f(x) = x2
Determine whether or not f is a bijection. If it is, prove it. If it is not, show why it is not.
b) Consider the set
F = {y | y = ax3 + b},
a, b being constants such that a \ne 0 and x \in R.
Is F equivalent to R? If so, prove it. If not, explain in details why it is not the case.a) Consider the whole of the English words set. Suppose an English word x is related to another
English word y if x and y begin with the same letter.
i) Show that this is an equivalence relation.
ii) Compute C(quadratic) and C(rhombus)
iii) How many equivalence classes are there in all, and why?
iv) What is the partition of the English words under this relation?
b) Consider Z, the set of integers. Suppose we define the relation: x is related to y if x - y > 3, x, y \in Z. Determine whether or not the relation isi) reflexive
ii) symmetric
iii) transitive
question 2
Consider the truth function
f(P, Q, R) = (P \rightarrow Q) \rightarrow Ra) Find a restricted statement form in conjunctive normal form (CNF) logically equivalent to f.
b) Find a restricted statement form in disjunctive normal form (DNF) logically equivalent to f.
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.
#339914 on Dec 2023