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Chapter 6: Q13. (page 211)

What can you conclude by using the given statement together with each additional statement? If no conclusion is possible, say so.

Given: If it is not raining, then I am happy.

a. I am not happy.

b. It is not raining.

c. I am overjoyed.

d. It is raining.

Short Answer

Expert verified

a. The conclusion is 鈥淚t is raining鈥�.

b.The conclusion is 鈥淚 am happy鈥�.

c.There is no conclusion.

d.The conclusion is 鈥淚 am not happy鈥�.

Step by step solution

01

a.Step 1 – Define the additional given statement

The given statement is 鈥淚f it is not raining, then I am happy鈥�

The given additional statement is 鈥淚 am not happy鈥�

02

– Define the contrapositive statement 

If the given statement is 鈥樷�橧f p then q鈥欌��, the contrapositive is 鈥樷�橧f not q then not p鈥欌赌�.

Since if it is not raining, then I am happy, therefore if I am not happy then it is raining.

03

– Conclusion for given statement together additional statement

Hence, the conclusion is 鈥淚t is raining鈥�.

04

b.Step 1 – Define the additional given statement

The given statement is 鈥淚f it is not raining, then I am happy鈥�

The given additional statement is 鈥淚t is not raining鈥�

05

– Explanation

If it is not raining, then I am happy.

06

– Conclusion for given statement together additional statement

Hence, the conclusion is 鈥淚 am happy鈥�.

07

c.Step 1 – Define the additional given statement

The given statement is 鈥淚f it is not raining, then I am happy鈥�

The given additional statement is 鈥淚 am overjoyed鈥�. As there is no relation between over enjoyment and

raining.

08

– Explanation

I am overjoyed. It is not related to the given statement.

09

– Conclusion for given statement together additional statement

Hence, there is no conclusion.

10

d.Step 1 – Define the additional given statement

The given statement is 鈥淚f it is not raining, then I am happy鈥�

The given additional statement is 鈥淚t is raining鈥�. As if it is not raining then I am happy therefore

if it is raining then I am not happy.

11

– Explanation

It is raining, then I am not happy.

12

– Conclusion for given statement together additional statement

Hence, the conclusion is 鈥淚 am not happy鈥�.

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