Split each of the following into simple sentences and determine whether it is true or false. Also, determine whether an ‘inclusive or’ or ‘exclusive or’ is used.
(i) The sum of 3 and 7 is 10 or $11 .$
(ii) $(1+i)$ is a real or a complex number.
(iii) Every quadratic equation has one or two real roots.
(iv) You are wet when it rains, or you are in a river.
(v) 24 is a multiple of 5 or $8 .$
(vi) Every integer is rational or irrational.
(vii) For getting a driving license, you should have a ration card or a passport. (viii) 100 is a multiple of 6 or $8 .$
(ix) Square of an integer is positive or negative.
(x) Sun rises or Moon sets.
(i) p: The sum of 3 and 7 is 10 or 11.
q: The sum of 3 and 7 is 10 .
r: The sum of 3 and 7 is 11 .
First sentence is TRUE. Second sentence is FALSE.
Or used is 'Exclusive or'.
(ii) $p:(1+i)$ is a real or a complex number.
$\mathrm{q}:(1+\mathrm{i})$ is a real number.
$r:(1+i)$ is a complex number.
First sentence is TRUE. Second sentence is FALSE.
Or used is 'Exclusive or'.
(iii) p: Every quadratic equation has one or two real roots.
q: Every quadratic equation has one real root.
r: Every quadratic equation has two real roots.
Both the sentences are FALSE. So, the compound sentence itself is FALSE.
(iv) p: You are wet when it rains, or you are in a river.
q: You are wet when it rains.
r: You are wet when you are in a river.
Both the component sentences are TRUE.
Or used is 'Inclusive or' because you can get wet either it rains or when you are in the river.
(v) p: 24 is a multiple of 5 or $8 .$
$\mathrm{q}: 24$ is a multiple of $5 .$
$\mathrm{r}: 24$ is a multiple of $8 .$
First sentence is FALSE. Second sentence is TRUE.
Or used is 'Exclusive or'.
(vi) p: Every integer is rational or irrational.
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q: Every integer is rational.
r: Every integer is irrational.
The first sentence is TRUE. The second sentence is TRUE. But both cannot be TRUE at the same time.
'Or' used is 'Exclusive or'.
(vii) p: For getting a driving license you should have a ration card or a passport.
q: For getting a driving license you should have a ration card.
r: For getting a driving license you should have a passport.
Both sentences are TRUE.
Or used is 'Inclusive or', because one can get a driving license with ration card or with passport or when they have both.
(viii) p: 100 is a multiple of 6 or $8 .$
q: 100 is a multiple of $6 .$
r: 100 is a multiple of 8 .
Both the sentences are FALSE. So the compound sentence itself is FALSE.
(ix) p: Square of an integer is positive or negative.
q: Square of an integer is positive.
r: Square of an integer is negative.
First sentence is TRUE. Second sentence is FALSE.
Or used is 'Exclusive or'.
(x) p: Sun rises or Moon sets.
q: Sun rises.
r: Moon sets.
Here both the sentences are TRUE, but only one occurs at a time.
So, Or used is 'Exclusive or'.