Exercise-2.1:
12. Determine all the truth value assignments if any for the primitive statement ,q,r,s,t that make each of the following compound statement FALSE.
a) [(p^q)^r] → ( s v t)
b) [p^(q^r)] → (s ⊻ t)
SOLUTION:
a) [(p^q)^r] → ( s v t)
This statement is false when [(p^q)^r]is TRUE and ( s v t) is FALSE.
Thus, P = q = r = True
And
Either s =t= False
Or S= True and t = False
Or S= False and t = True.
b) [p^(q^r)] → (s ⊻ t)
This statement is false when [p^(q^r)] is TRUE and (s ⊻ t) is FALSE.
Thus,
P = q = r = TRUE And s = t = TRUE
Or
P = q = r = TRUE And s = t = FALSE
Exercise-2.2:
12. Show that for primitive statement p,q
p ⊻ q ⇔ [ ( p ^ ~q) v ( ~p ^q ) ⇔ ~(p ↔ q )
SOLUTION:
Consider [ ( p ^ q ) v ( ~ p ^ q ) ]
Now,
[ ( p ^ q ) v ( ~ p ^ q ) ]
=[ ~(p→q) v( q ^ ~p ) ] -Commutative law & ~(p→q)⇔(q^~p)
=[~(p→q) v ~(q→p)] - ~(p→q)⇔(q^~p)
=~[( p→q)^ (q→p)] -Demorgan’s Law
=~(p ↔ q ) - ( p→q)^ (q→p)⇔(p ↔ q )
= p ⊻ q - (p ↔ q ) ⇔~(p ⊻ q)
Exercise-2.4:
5. Proffessor Carlson’s class in mechanics is comprised of 29 students of which exactly
1) Three physics majors are juniors.
2) Two electrical engineering majors are juniors.
3) Four mathemaatics maajors are juniors.
4) Twelve physics majors are seniors.
5) Four electrical engineering majors are seniors.
6) Two electrical engineering majors are graduate students ; and
7) Two mathematics majors are graduate students.
Consider the following open statements.
c(x): Student x is in the class ( that is , Proffessor Carlson’s mechanics as already described ).
j(x): Student x is a junior.
s(x): Student x is a senior.
g(x): Student x is a graduate student.
p(x): Student x is a physics major.
e(x): Student x is an electrical engineering major.
m(x): Student x is a mathematics major.
Write each of the following statements in terms of quantifiers and the open statements c(x),j(x),s(x),g(x),p(x),e(x) and m(x) , and determine whether the given statement is ttrue or false. Here the universe comprise of all 12,500 students enrolled at the universitywhere Proffessor Carlson teaches. Furthermore, at this university each studentt haas only one major.
a) There is a mathemtics major in the class who is a junior.
b) There is a senior in the class who is not a mathematics major.
c) Every student in the class is majoring in mathematics.
d) No graduate student in the class is a physics maajor.
e) Every senior in the class is majoring in either physics or electrical engineering.
SOLUTION:
a) ∃x [m(x) ^ c(x) ^j(x) ] ------------------------TRUE
b) ∃x[s(x)^c(x)^~m(x)] --------------------------TRUE
c) ∀x[c(x) →(m(x) ⊻ p(x))] --------------------------FALSE
d) ∀x[(g(x) ^c(x)) →~p(x)]--------------------------TRUE
e) ∀x[(c(x) ^s(x)) →(p(x) ⊻ e(x))]-----------------TRUE
Exercise-3.1:
12:
Let S = {1,2,3,…………………………,29,30}.
How many subsets A of S satisfy
a) |A|=5 ?
b) |A|=5 and the smallest element in A is 5 ?
c) |A|=5 and thee smallest element in A is less than 5 ?
SOLUTION:
a) 5 elements can be selected from 30 elements in 30C5 ways.
Thus, 30C5 subsets of S satisfy the given condition.
b) S – {1,2,3,4} ={5,6,7………………….,29,30} = B
|B| = 26
Sincce 5 is the smallest number in each case,it is required to select 4 numbers from the remaining 25 numbers.
Thus the 4 numbers can bbe selected in 25C4 ways.
Thus, 25C4 subsets satisfy the given condition.
c) Here the smallest element can be 1,2,3 or 4.
S-{5} = B
|B|=29
Now,
Case1:
Let us consider smallest element is 1.
Thus we need 4 more elements for the required subset.
It can be done in 29C4 ways.
Case2:
Let us consider smallest elemnt is 2.
Here 1 has to be omitted from B, since smallest number in this case is 2..
Thus remaining 4 number can be selected in 28C4 ways.
Case 3:
Let us consider the smallest number is 3.
Thus the remaining 4 number can be selected in 27C4 ways.
Case4:
Let us consider the smallest number is 4.
Thus the remaining 4 numbers can be selected in 26C4ways.
Thus , total number of ways is=
Thus , total number of ways is=
29C4 +28C4 + 27C4 + 26C4ways.
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