1)Verify that
[p->q->r)] -> [(p->q)->(p->r)] is a tautology.
Soln:
P q r [p->(q->r)] [(p->q)->(p->r)] [p->(q->r)]->[(p->q)->(p->r)]
1 1 1 1 1 1
1 1 0 0 0 1
1 0 1 1 1 1
1 0 0 1 1 1
0 1 1 1 1 1
0 1 0 1 1 1
0 0 1 1 1 1
0 0 0 1 1 1
From the above truth table it is verified it is TAUTOLOGY.
1) Determine whether each of the following is TRUE or FALSE.Here p , q are arbitrary statements.
a) An equivalent way to express the converse of “p is sufficient for q” is “p is necessary for q”.
b) An equivalent way to express the inverse of “p is necessary for q” is “~q is sufficient for ~p”.
c) An equivalent way to express the contrapositive of “p is necessary for q” is “~q is necessary for ~p”.
Soln: As we know pàq means that------------
p is a sufficient condition for q
q is necessary condition for p
q is necessary for p
p is sufficient for q
a)TRUE
p is sufficient for q p->q
p is necessary for q q->p which is the converse of p->q.
b)TRUE
p is necessary for q q->p
~q is sufficient for ~p ~p-> ~q which is inverse of p->q
c)TRUE
p is necessary for q q->p
~q is necessary for ~p ~q-> ~p which is contra positive of pàq
2) Let P(x) be the open statement “ x2 = 2x”,where the universe comprises all integers.Determine whether each of the following statement is TRUE or FALSE.
a) P(0)
P(x): x2 = 2x
P(0): 0 = 2*0
P(0):0 = 0
Hence TRUE
b) P(1)
P(x):x2 = 2x
P(1):1=2
FALSE
c) P(2)
P(x):x2 = 2x
P(2):4=4
TRUE
d) P(-2)
P(x):x2 = 2x
P(x):4=-4
FALSE
e)For some x p(x)
p(x):x2 =2x
Put x=0
0 = 0 for some P(x) is true.
TRUE
f) For all x, p(x)
Put x=3 belongs to Z
Hence P(x) becomes
(3)2= 2*3
9≠6
FALSE
4)which of the following sets are nonempty?
a) {x|x ϵ N,2x+7=3}
consider: 2x+7=3;
2x =3-7
x=-2
since N is a natural number here no value of x satisfy the above condition.
Hence EMPTY SET.
b) {x ϵ Z | 3x + 5 =9}
consider: 3x + 5 =9;
x = 4/3
since Z is set of integers here no value of x which satisfy the above condition which is integer.Hence EMPTY SET.
c) {x ϵ Q ,x2 + 4 = 6}
x2 + 4 =6
X2=6 – 4
X2 = 2
X =sqrt 2.
Since Q is set of rational nos and sqrt 2 is a irrational no.
Hence EMPTY SET.
d) {x ϵ R | x2 + 4 = 6 }
According to previous example
X = sqrt 2.
Since R is set of real nos and sqrt 2 is a real no.
Hence NON EMPTY SET.
e) {x ϵ R | x2 + 3x +3 = 0}
X2 + 3X + 3 = 0
X = complex roots
Since R is set of real nos.
Hence EMPTY SET.
f) {x ϵ C | x2 + 3x +3 = 0}
X2 + 3X + 3 = 0
X2 + 3X + 3 = 0
X = complex roots
Since C is set of complex nos.
Hence NON EMPTY SET.
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