Search & Filtering
List the elements of {1, 2, 3, 4} ∩ {2, 3, 5, 7}
3 + 5 = 8 is a proposition for p, find the corresponding ῀p.
prove a --> ( b V c ) using contradiction method and combination of inference rules and equivalence laws from these premises : 1. a --> ( d V b ) 2. d --> c
R1={(a, b) € R |a >b|}, the "greater than relation.
R2=(a, b) € R|a>=b|}. the "greater than or equal to relation
R3=(a, b) €R |a < b), the "less than" relation.
R4=(a, b). the "less than or equal to
R5=(a , b) € R |a = b|. the "equal to" relation
In the grammar of natural languages, such as Sesotho, noun words are categorised into equivalent noun classes, based on their generating “stems”. A stem is a noun word prefix, which is the “first syllable” of the noun word, from where suffixes could be attached to generate different noun words. In Sesotho, in particular, the equivalence noun classes are based on the following two relations:
R1: A noun word x is related to another noun word y iff x and y have the same stem
R2: A noun word x is related to another noun word y iff x and y have no stems. R2
generates the equivalence Class 9.
a) Prove that:
i) R1 is an equivalence relation
ii) R2 is also an equivalence relation
b)
i) List any three (3) noun words belonging to Class 9
ii) Show that “se” is a stem by showing that C(“sekatana”) is an equivalence class by listing any first eleven (11) words in it.
a) Write a finite state machine (FSM) for an automatic reversible 6 modulo counter as follows:
The counter counts 0, 1, 2, 3, 4, 5 (when its internal memory input is registering 0) and reverses: 5 4, 3, 2, 1, 0 (when its internal memory input is registering 1)
b) As a result of a) above, design and implement a minimal logic circuit for the
automatic reversible 6 modulo counter. If the counter were to enter any of the
unwanted states, it should be reset to count from 0 on the next clock pulse.
Hint: The counter uses 4 bits. Let the most significant bit (MSB) be the memory bit
that the counter will set as 0 for counting up and will reset it to 1 when counting
downwards.
a) Find a restricted statement form in conjunctive normal form (CNF) logically
equivalent to the statement form:
¬P Q "\\land" ¬R
b) Find a restricted statement form in disjunctive normal form (DNF) logically
equivalent to the statement form:
¬P Q "\\land" ¬R
a) Let \mathbb{N} be the set of natural numbers, O, the set of odd numbers and S, the set ofsquare numbers. Consider the bijections:
f : \mathbb{N} "\\to" O, f(n) = 2n + 1
g : O S, g(d)=\frac{d2 -2d +1}{4}
Consider f \circ g.Determine whether or not the specified composition is possible. If it is not possible,
explain in details why it is not. If it is possible, then
i) compute:
1. the resulting composition
2. the corresponding image set
ii) Determine whether or not the specified composition is a bijection.
Design and implement a minimal 5 modulo up counter. It counts from 0 to 4 and repeats.
Design the circuit such that, if the counter enters into the unwanted states: 5, 6 and 7, it should jump into state 0 on the next clock pulse.
Consider the human race. Suppose a person x is related to another human being y if x
has the same father as y.
a) Prove that this is an equivalence relation.
b) Compute C(Thabane) and C(Mosisili)
c) Can C(Thabane) and C(Mosisili) ever intersect? If so, how so. If not, why not.
d) Compute the partition of the human race under this relation.
