1. Determine the truth value of each of the fallowing implications.
a).if 3+4=12,then 3+2=6.
b).if 3+3=6,then3+4=9
c).if thomas jefferson was the third president of united
stases ,then 2+3=5.
Ans. Let the statements
a). P=3+4=12, then q=3+2=6
p→q
true→false=false.
b).let p=3+3=6, q=3+4=9
p→q
true→false=false.
c).p=above statement ,q=2+3=5,
if first statement is true then second staments is
always true so the result is true.
If first statement is not true then also result is true.
2).Negate each of the fallowing and simplyfy the statement
a).p^(q^r)^(~pv~qvr)
b).p^q)→r
c).p→(~q^r)
d).pvqv(~p^~q^r)
Ans.
a. P^(qvr) ^(~pv~qvr)
~{pΛ(qvr)} ^~{(~pv~qvr)}
~pv~q^~r^p^q^~r
b. (p^q)→r
~(p^q)vr
~{~(p^q)vr}
P^q^~r
c. p→(~q^r)
~pv(~q^r)
~{~pv(~q^r)}
P^qv~r
d. pvqv(~p^~q^r)
~{pvqv(~p^~q^r)}
~p^~q^pvqv~r
3.prove by rule inference
[p^(q^r)] v ~[pv(q^r)]
[p^(qvr)] v ~[pv(qvr)]
q^r
---------------
qvr
steps reasons
[p^q(q^r)]v~[pv(q^r)] premises
[p^(qvr)]v~[pv(qvr)] premises
q^r premises
qvr by step 1,3 and by rule of proof by cases
4.consider a fallowing subsets of Z.
A={2m+1/m€Z}
B={2n+3/n€Z}
C={2p-3/p€Z}
D={3r+1/r€Z}
E={3s+2/s€Z}
F={3t-2/t€Z}
Findout which are the fallowing are true.
1.A=B 2.A=C 3.B=C 4.D=E 5.D=F 6.E=F
Ans.
A={2m+1/m€Z} A={-5,-3,-1,1,0,3,5,7}
B={2n+3/n€Z} B={-9,-7,-5,-3,-1,1,3,5,7}
C={2p-3/p€Z} C={-9,-7,-5,-3,-1,1,3,5,7,9}
D={3r+1/r€Z} D={-2,-5,-8,1,4,7,10,13}
E={3s+2/s€Z} E={-1,-4,-7,2,5,8,11}
F={3t-2/t€Z} F={-5,-8,-11,1,4,7}
1.A=B true
2.A=C true
3.B=C true
4.D=E false
5.D=F true
6.E=F false
-----------------The end------------------
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