Split each of the following into simple sentences and determine whether it is true or false.
(i) A line is straight and extends indefinitely in both the directions.
(ii) A point occupies a position, and its location can be determined.
(iii) The sand heats up quickly in the sun and does not cool down fast at night.
(iv) 32 is divisible by 8 and 12.
(v) x = 1 and x = 2 are the roots of the equation x2 – x – 2 = 0.
(vi) 3 is rational, and √3 is irrational.
(vii) All integers are rational numbers, and all rational numbers are not real numbers.
(viii) Lucknow is in Uttar Pradesh, and Kanpur is in Uttarakhand.
p: A line is straight.
q: A line extends indefinitely in both the directions.
Both the simple sentences are TRUE; therefore, the given sentence is TRUE.
(ii) p: A point occupies a position.
q: Its location can be determined.
Both the simple sentences are TRUE; therefore, the given sentence is TRUE.
(iii) p: The sand heats up quickly in the sun.
q: The sand does not cool down fast at night.
The first sentence is TRUE and the second sentence is FALSE.
Therefore the given sentence is FALSE.
(iv) p: 32 is divisible by 8.
q: 32 is divisible by 12.
First sentence is TRUE. Second sentence is FALSE.
Therefore the given sentence is FALSE.
(v) p: x = 1 is a root of the equation x2 – x – 2 = 0
q: x = 2 is a root of the equation x2 – x – 2 = 0
So, the first sentence is FALSE. Second sentence is TRUE.
Therefore, the given sentence is FALSE.
(vi) p: 3 is rational.
q: √3 is irrational.
First sentence is TRUE. Second sentence is TRUE.
Both the sentences are TRUE; therefore the given sentence is TRUE.
(vii) p: All integers are rational numbers.
q: All rational numbers are not real numbers.
First sentence is TRUE. Second sentence is FALSE.
Therefore, the given sentence is FALSE.
(viii) p: Lucknow is in Uttar Pradesh.
q: Kanpur is in Uttarakhand.
First sentence is TRUE. Second sentence is FALSE.
Therefore, the given sentence is FALSE.