In the previous section, we saw validating statements. In this section, we will see some miscellaneous examples.
Solved example 14.14
Check whether “Or” used in the following compound statement is exclusive or inclusive. Write the component statements of the compound statement and use them to check whether the compound statement is true or not. Justify your answer.
t: you are wet when it rains or you are in a river.
Solution:
1. We have:
t: you are wet when it rains or you are in a river.
2. The component statements are:
p: you are wet when it rains.
q: you are wet when you are in a river.
♦ p and q are connected by “or”.
3. Let us analyze the component statements:
•
During rain, you are wet.
•
When in a river, you are wet.
•
When in a river, if it rains, you are wet.
•
So the “or” used in this case is inclusive “or”.
4. Both the component statements are true. So t is true.
Solved example 14.15
Write the negation of the following statements
(i) p: For every real number x, x2 > x.
(ii) q: There exists a rational number x such that x2 = 2.
(iii) r: All birds have wings.
(iv) s: All students study mathematics at the elementary level.
Solution:
Part (i)
1. Consider the statement:
p: For every real number x, x2 > x.
2. We want ~p.
• p tells us that: For all real numbers, the given property is valid.
• ~p must tell us that "For all" is false.
3. This ~p can be achieved by saying that there is at least one real number for which the property is not valid.
• So we get:
~p: There exists one real number x for which x2 ≤ x.
Part (ii)
1. Consider the statement:
q: There exists a rational number x such that x2 = 2.
2. We want ~q.
• q tells us that: There is at least one rational number for which the given property is valid.
• ~q must tell us that "at least one" is false.
3. This ~q can be achieved by saying that for all rational numbers, the property is not valid.
• So we get:
~p: For all rational numbers x, x2 ≠2.
Part (iii)
1. Consider the statement:
r: All birds have wings.
2. We want ~r.
• r tells us that: For all birds, the given property is valid.
• ~r must tell us that "For all" is false.
3. This ~r can be achieved by saying that there is at least one bird for which the property is not valid.
• So we get:
~r: There exists one bird which do not have wings.
Part (iv)
1. Consider the statement:
s: All students study mathematics at the elementary level.
2. We want ~s.
• s tells us that: For all students, the given property is valid.
• ~s must tell us that "For all" is false.
3. This ~r can be achieved by saying that there is at least one student for which the property is not valid.
• So we get:
~r: There exists one student who does not study mathematics at the elementary level.
Solved example 14.16
Using the words “necessary and sufficient”, rewrite the statement “The integer n is odd if and only if n2 is odd”. Also check whether the statement is true.
Solution:
Part (i): Rewriting the statement
Using the words “necessary and sufficient”, we can write:
“The integer n is odd” is necessary and sufficient condition for “n2 to be odd” and vice versa.
Part (ii): Checking the validity.
• The given compound statement is in the form “p if and only if q”.
The component statements are:
p: The integer n is odd.
q: n2 is odd.
• In such cases, we know that:
♦ Whenever p is true, q is also true.
♦ Whenever q is true, p is also true.
• So we need to show two items:
Case (i) If p is true, then q is true.
Case (ii) If q is true, then p is true.
Case (i): If p is true, then q is true.
♦ First, we assume that p is true.
♦ Based on this assumption, we check q.
1. Assuming p to be true:
• Let us assume that, integer n is indeed odd.
2. Since it is an odd integer, we can write:
♦ n = 2m + 1
✰ Where m is some integer.
3. Now we calculate n2. We get:
$\begin{array}{ll}
{}&{n^2} & {~=~}& {(2m+1)^2} &{} \\
{}&{} & {~=~}& {4m^2 + 4m + 1} &{} \\
{}&{} & {~=~}& {2(2m^2 + 2m) + 1} &{} \\
\end{array}$
4. Consider the result in (3):
• m is an integer.
⇒ m2 will be an integer.
⇒ (2m2 + 2m) will be even.
⇒ 2(2m2 + 2m) will be even.
⇒ 2(2m2 + 2m) + 1 will be odd.
5. Based on (3) and (4), we can write:
n2 is odd.
• So the statement q is true.
6. We obtained the truth value of q as T.
• We obtained this by assuming that the truth value of p is T.
• So the truth value of “if p then q” is T.
Case (ii): If q is true, then p is true.
Here we will use the contrapositive method. That is., if ~p then ~q.
♦ First, we assume that ~p is true.
♦ Based on this assumption, we check ~q.
The statements are:
~p: The integer n is even.
~q: n2 is even.
1. Assuming ~p to be true:
• Let us assume that, n is an even integer and n2 is indeed even.
2. Since it is an even integer, we can write:
♦ n = 2m
✰ Where m is some integer.
3. Now we calculate n2. We get:
$\begin{array}{ll}
{}&{n^2} & {~=~}& {(2m)^2} &{} \\
{}&{} & {~=~}& {4m^2} &{} \\
{}&{} & {~=~}& {2(2m^2)} &{} \\
\end{array}$
4. Consider the result in (3):
• m is an integer.
⇒ m2 will be an integer.
⇒ 2m2 will be even.
⇒ 2(2m2) will also be even.
5. Based on (3) and (4), we can write:
n2 is even.
• So the statement ~q is true.
6. We obtained the truth value of ~q as T.
• We obtained this by assuming that the truth value of ~p is T.
• So the truth value of “if ~p then ~q” is T.
• So the truth value of “if q then p” is T.
◼ From case (i), we obtained:
• The truth value of “if p then q” is T.
◼ From case (ii), we obtained:
• The truth value of “if q then p” is T.
◼ So we can write: The given "p if and only if q" is true.
•
That is.,
The statement “The integer n is odd if and only if n2 is odd” is true.
Solved example 14.17
For the given statement, identify the necessary and sufficient conditions
t: If you drive over 80 km per hour, then you will get a fine.
Solution:
1. Consider the statement:
t: If you drive over 80 km per hour, then you will get a fine.
2. The component statements are:
p: You drive over 80 km per hour.
q: You will get a fine.
3. This is a statement with "if p then q".
•
So we can write:
♦ p is a sufficient condition for q.
♦ q is a necessary condition for p.
(see example 2 at the beginning of section 14.3)
4. Thus we get:
•
The sufficient condition is:
Driving over 80 km per hour.
(This is sufficient to get a fine)
•
The necessary condition is:
Getting a fine.
(This is necessary if the speed is above 80 km per hour)
The link below gives some more solved examples:
In the next chapter, we will see statistics.
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