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(i)
The components of the compound statement:
The sky is blue and the grass is green.
are :
p : The sky is blue
q : The grass is green
(ii)
The earth is round or the sun is cold.
are:
p : The earth is round
q : The sun is cold.
(iii)
All rational numbers are real and all real numbers are complex.
p : All rational numbers are real
q: All real numbers are complex
(iv)
25 is a multiple of and 8.
p :25 is multiple of 5
q: 25 is multiple of 8.
In the statement:
Students can take Hindi or Sanskrit as their third language.
an exclusive "OR" is used because
A student cannot take both Hindi and Sanskrit as the third language.
To entry a country, you need a passport or a voter registration card.
an inclusive "OR' is used because
Since a person can have both a passport and a voter registration card to enter a country.
A lady gives birth to a baby boy or a baby girl.
A lady cannot give birth to a baby who is both a boy and a girl.
To apply for a driving licence, you should have a ration card or a passport.
an inclusive OR is used because
A person can have both a ration card and passport to apply for a driving licence.
The component statements of the compound statement:
To enter into a public library children need an identity card from the school or a letter from the school authorities.
are
p : To get into a public library children need an identity card.
q : To get into a public library children need a letter from the school authorities.
We know that if p and q are true then p or q must also be true. Hence, the compound statement is true.
All rational numbers are real and all real numbers are not complex.
p: All rational numbers are real.
q : All real numbers are not complex.
we know that p is true and q is false.
∴ The compound statement “p and q” is false.
Square of an integer is positive or negative.
p : Square of an integer is positive.
q: Square of an integer is negative.
We know that if p and q are true then p or q must also be true.
Hence, the compound statement is true.
x = 2 and x = 3 are the roots of the equation 3x2 - x -10 = O.
p: x - 2 is a root of the equation 3x2 – x - 10 = 0.
q: x = 3 is a root of the equation 3x2 - x - 10 = O.
Here, p is true but q is false.
∴ The compound statement "p and q° is false.
(v)
The sand heats up quickly in the sun and does not cool down fast at night.
p: The sand heats up quickly in the sun.
q: The sand does not cool down fast at night.
The compound statement p and q is false, because p is true and q is false.
Delhi is in India and 2 + 2 = 4.
p : Delhi is in India.
q : 2 + 2 = 4.
Delhi is in England and 2 + 2 = 4.
p : Delhi is in England
Here p is false and q is true. So, p and q must be false. Hence, the compound statement is false.
Delhi is in India and 2 + 2 = 5.
p: Delhi is in India
q : 2 + 2 = 5.
Here, p is true and q is false. So, p and q must be false. Hence, the compound statement is false.
Delhi is in England and 2 + 2 = 5.
p: Delhi is in England.
We know that if p and q are false then p and q must also be false. Hence, the compound statement is false.
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Chapter 31: Mathematical Reasoning –...