Question

Write the component statements of the following compound statements and check whether the compound statement is true or false:

(i) To enter into a public library children need an identity card from the school or a letter from the school authorities.

(ii) All rational numbers are real and all real numbers are not complex.

(iii) Square of an integer is positive or negative.

(iv) x = 2 and x = 3 are the roots or the equation 3x^{2} − x − 10 = 0.

(v) The sand heats up quickly in the sun and does not cool down fast at night.

(i) To enter into a public library children need an identity card from the school or a letter from the school authorities.

(ii) All rational numbers are real and all real numbers are not complex.

(iii) Square of an integer is positive or negative.

(iv) x = 2 and x = 3 are the roots or the equation 3x

(v) The sand heats up quickly in the sun and does not cool down fast at night.

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Solution

(i) The component statements of the given compound statement are:

1) To enter into a public library, children need an identity card from the school.

2) To enter into a public library, children need a letter from the school authorities.

The compound statement is true because both component statements are true.

(ii) The component statements of the given compound statement are:

1) All rational numbers are real.

2) All real numbers are not complex.

The compound statement is false because all real numbers are complex. The connective used is "and". So, even if one component statement is false, the compound statement is false.

(iii) The component statements of the given compound statement are:

1) Square of an integer is positive.

2) Square of an integer is negative.

The compound statement is true because the first statement is true. Since the connective used is "or" and one of the component statements is true, the compound statement is true.

(iv) The component statements of the given compound statement are:

1) $x=2$ is the root or the equation $3{x}^{2}-x-10=0$.

2) $x=3$ is the root or the equation $3{x}^{2}-x-10=0$.

The connective used is "and". So, both component statements must be true for the compound statement to be true. The statement "$x=3$ is the root or the equation $3{x}^{2}-x-10=0$" is false. Therefore, the compound statement is false.

(v) The component statements of the given compound statement are:

1) The sand heats up quickly in the sun.

2) Sand does not cool down fast at night.

The compound statement uses "and" as the connective. For the compound statement to be true, both the component statements must be true. The second component statement "Sand does not cool down fast at night" is false. Sand cools down fast at night. Therefore, the compound statement is false.

1) To enter into a public library, children need an identity card from the school.

2) To enter into a public library, children need a letter from the school authorities.

The compound statement is true because both component statements are true.

(ii) The component statements of the given compound statement are:

1) All rational numbers are real.

2) All real numbers are not complex.

The compound statement is false because all real numbers are complex. The connective used is "and". So, even if one component statement is false, the compound statement is false.

(iii) The component statements of the given compound statement are:

1) Square of an integer is positive.

2) Square of an integer is negative.

The compound statement is true because the first statement is true. Since the connective used is "or" and one of the component statements is true, the compound statement is true.

(iv) The component statements of the given compound statement are:

1) $x=2$ is the root or the equation $3{x}^{2}-x-10=0$.

2) $x=3$ is the root or the equation $3{x}^{2}-x-10=0$.

The connective used is "and". So, both component statements must be true for the compound statement to be true. The statement "$x=3$ is the root or the equation $3{x}^{2}-x-10=0$" is false. Therefore, the compound statement is false.

(v) The component statements of the given compound statement are:

1) The sand heats up quickly in the sun.

2) Sand does not cool down fast at night.

The compound statement uses "and" as the connective. For the compound statement to be true, both the component statements must be true. The second component statement "Sand does not cool down fast at night" is false. Sand cools down fast at night. Therefore, the compound statement is false.

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