Question

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

a. The conclusion is “It is raining”.

b.The conclusion is “I am happy”.

c.There is no conclusion.

d.The conclusion is “I am not happy”.

Step-by-step solution

a.Step 1 – Define the additional given statement

The given statement is “If it is not raining, then I am happy”

The given additional statement is “I am not happy”

– Define the contrapositive statement 

If the given statement is ‘’If p then q’’, the contrapositive is ‘’If not q then not p’’.

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

– Conclusion for given statement together additional statement

Hence, the conclusion is “It is raining”.

b.Step 1 – Define the additional given statement

The given statement is “If it is not raining, then I am happy”

The given additional statement is “It is not raining”

– Explanation

If it is not raining, then I am happy.

– Conclusion for given statement together additional statement

Hence, the conclusion is “I am happy”.

c.Step 1 – Define the additional given statement

The given statement is “If it is not raining, then I am happy”

The given additional statement is “I am overjoyed”. As there is no relation between over enjoyment and

raining.

– Explanation

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

– Conclusion for given statement together additional statement

Hence, there is no conclusion.

d.Step 1 – Define the additional given statement

The given statement is “If it is not raining, then I am happy”

The given additional statement is “It is raining”. As if it is not raining then I am happy therefore

if it is raining then I am not happy.

– Explanation

It is raining, then I am not happy.

– Conclusion for given statement together additional statement

Hence, the conclusion is “I am not happy”.