1. Express the assertions given below as a proposition of a predicate logic using
the predicates indicated below. The universe is the set of objects.[15]
C(x): x is a car.
E(x): x is expensive.
S(x): x is slow.
T(x): x is a train.
ST(x,y ): x is slower than y: y is faster than x.
(a) Some cars are slow but not expensive.
(b) Only cars are slow.
(c) A car is not expensive only if it is slow.
(d) Some trains are slower than any car.
(e) Some expensive cars are faster than some trains.
2 (a) Prove that
.
[10]
(b) Is A equal to B if
A - B = B - A ? Justify your answer. [10]
3. Test the following binary relation R on the given set S for reflexivity,
irreflexivity, antisymmetry, symmetry and transitivity. Fill in the table below. [15]
(a) S is the set of real numbers.
iff
x2 - y2 = 0.
(b) S is a collection of sets.
iff A - B = .
(c) S is the set of real numbers.
iff x*y is an even number.
(d) S is a collection of sets.
iff
.
(e) S is a set of people.
iff x and y take some courses together.
Question | Reflexive | Irreflexive | Antisymmetric | Symmetric | Transitive |
(a) | |||||
(b) | |||||
(c) | |||||
(d) | |||||
(e) |
4. Prove the following by mathematical induction:
(a)
02 + 22 + 42 + ... + (2n)2 = 2n( n + 1 )( 2n + 1 )/3. [10]
(b) n3 - n is an even number if .
[10]
5. Let R be the relation on the set of propositions defined as follows:
if and only if
.
Answer the following questions:
(a) Prove that R is an equivalence relation. [9]
(b) Give two different examples of equivalence class of R. [6]
(c) What are the members of the equivalence class [True] ? [3]
(d) Let X be a symbolic logical form for a proposition in English (such as
).
Which of the following eight propositions are in the same equivalence class as X ? [12]
,
,
(
),
,
,
,
and
.