In the previous section, we saw compound statements and component statements. In this section, we will see Special words/phrases.
Special words/phrases
•
We saw that, component statements are connected by words like “and”, “or” etc.,
•
These words are known as connectives.
•
When used in mathematical statements, these words have special meanings. We need to clearly understand those special meanings.
The word “And”
This can be explained in 3 steps:
1. First we will see an example. It can be written in 4 steps:
(i) Given below, is a compound statement with the word “and”:
p: A point occupies a position and it’s location can be determined.
(ii) The component statements are:
q: A point occupies a position.
r: It’s location can be determined.
♦ q and r are connected by “and”.
(iii) Checking component statements:
•
q is true.
•
r is true.
(iv) Since both q and r are true, p is true.
2. Let us see another example. It can be written in 4 steps:
(i) Given below, is a compound statement with the word “and”:
p: 42 is divisible by 5, 6 and 7.
(ii) The component statements are:
q: 42 is divisible by 5.
r: 42 is divisible by 6.
s: 42 is divisible by 7.
♦ q, r and s are connected by “and”.
(iii) Checking component statements:
•
q is false.
•
r is true.
•
s is true
(iv) Since q is false, p is false.
(Even though r and s are true, p is false because one component q is false)
3. We can write two rules related to “and”:
(i) A compound statement with “and” is true if all it’s component statements are true.
(ii) A compound statement with “and” is false if one or more of it’s component statements is false.
Let us see a solved example
Solved example 14.5
Write the component statements of the following compound statements and check whether the compound statements are true or false.
(i) A line is straight and extends indefinitely in both directions.
(ii) 0 is less than every positive integer and every negative integer.
(iii) All living things have two legs and two eyes.
Solution:
Part (i):
1. We have:
p: A line is straight and extends indefinitely in both directions.
2. The component statements are:
q: A line is straight.
r: A line extends indefinitely in both directions.
♦ q and r are connected by “and”.
3. Checking component statements:
•
q is true
•
r is true.
4. Since both q and r are true, p is true.
Part (ii):
1. We have:
p: 0 is less than every positive integer and every negative integer.
2. The component statements are:
q: 0 is less than every positive integer.
r: 0 is less than every negative integer.
♦ q and r are connected by “and”.
3. Checking component statements:
•
q is true
•
r is false.
4. Since r is false, p is false.
Part (iii):
1. We have:
p: All living things have two legs and two eyes.
2. The component statements are:
q: All living things have two legs.
r: All living things have two eyes.
♦ q and r are connected by “and”.
3. Checking component statements:
•
q is false
•
r is false.
4. Since one or more of the component statements is false, p is false. In this case both the components q and r are false.
Let us see an interesting case involving “and”. It can be written in 4 steps:
1. Consider the sentence:
A mixture of alcohol and water can be separated by chemical methods.
2. We are tempted to write two separate sentences:
(i) A mixture of alcohol can be separated by chemical methods.
(ii) A mixture of water can be separated by chemical methods.
3. Let us analyze the two sentences:
(i) The sentence in 2(i) does not have any meaning. This is because, if alcohol alone is present, we cannot call it a ‘mixture’. If there is no ‘mixture’, there is no need for separating.
(ii) The sentence in 2(ii) does not have any meaning. This is because, if water alone is present, we cannot call it a ‘mixture’. If there is no ‘mixture’, there is no need for separating.
4. It is clear that, the word “and” in this case is not used for connecting two statements. It is used for connecting two substances alcohol and water.
The word “Or”
This can be explained in 6 steps:
1. First we will see an example. It can be written in 4 steps:
(i) Given below, is a compound statement with the word “or”:
p: Two lines in a plane, either intersect at a point or they are parallel.
(ii) The component statements are:
q: Two lines in a plane, intersect at a point.
r: Two lines in a plane, are parallel.
♦ q and r are connected by “or”.
(iii) Checking component statements:
•
if q is true, then r is false.
•
if r is true, then q is false.
(iv) One of the components will be always true.
If one of the components (q or r) is true, p is true.
2. Let us see another example. It can be written in 3 steps:
(i) Given below, is another compound statement with the word “or”:
p: 125 is a multiple of 7 or 8.
(ii) The component statements are:
q: 125 is a multiple of 7.
r: 125 is a multiple of 8.
♦ q and r are connected by “or”.
(iii) Checking component statements:
•
q is false.
•
r is false.
(iv) If all the components are false, p is false.
3. Let us see one more example. It can be written in 3 steps:
(i) Given below, is a compound statement with the word “or”:
p: The school is closed if there is a holiday or sunday.
(ii) The component statements are:
q: The school is closed if there is a holiday.
r: The school is closed if it is sunday.
♦ q and r are connected by “or”.
(iii) Checking component statements:
•
q is true.
•
r is true.
(iv) If one of the components is true, p is true.
4. We can write two rules related to “or”:
(i) A compound statement with “or" is true if one or more of it’s component statements are true.
(ii) A compound statement with “or” is false if all of it’s component statements are false.
5. The word "or" is used in two ways in English language. This can be explained using two examples.
Example 1:
This can be written in 3 steps:
(i) Consider the compound statement with word "or":
p: An ice cream or pepsi is available with lunch in a restaurant.
(ii) The component statements are:
q: An ice cream is available with lunch in a restaurant.
r: A pepsi is available with lunch in a restaurant.
♦ q and r are connected by “or”.
(iii) Let us analyze the component statements:
•
If a customer does not want pepsi, he can opt for ice cream.
•
If a customer does not want ice cream, he can opt for pepsi.
•
Only one item (ice cream or pepsi) will be allowed.
•
The “or” used in such cases is called exclusive “or”.
Example 2:
This can be written in 3 steps:
(i) Given below, is a compound statement with the word “or”:
p: A student who has taken biology or chemistry can apply for M.Sc. microbiology program.
(ii) The component statements are:
q: A student who has taken biology can apply for M.Sc. microbiology program.
r: A student who has taken chemistry can apply for M.Sc. microbiology program.
♦ q and r are connected by “or”.
(iii) Let us analyze the component statements:
•
If a student has taken biology, he/she can apply for M.Sc. microbiology program.
•
If a student has taken chemistry, he/she can apply for M.Sc. microbiology program.
•
If a student has taken both biology and chemistry, he/she can apply for M.Sc. microbiology program.
•
The “or” used in such cases is called inclusive “or”.
6. We must clearly understand the difference between exclusive “or” and inclusive “or”. Then only we will be able to check whether a compound statement is true or false.
Now we will see some solved examples.
Solved example 14.6
For each of the following statements, determine whether an inclusive “or” or exclusive “or” is used. Give reasons for your answer.
(i) To enter a country, you need a passport or a voter registration card.
(ii) The school is closed if it is a holiday or a Sunday.
(iii) Two lines intersect at a point or are parallel.
(iv) Students can take French or Sanskrit as their third language.
Solution:
Part (i):
1. We have:
p: To enter a country, you need a passport or a voter registration card.
2. The component statements are:
q: To enter a country, you need a passport.
r: To enter a country, you need a voter registration card.
♦ q and r are connected by “or”.
3. Let us analyze the component statements:
•
If a person has a passport, he/she can enter a country.
•
If a person has a voter registration card, he/she can enter a country.
•
If a person has both passport and voter registration card, he/she can enter a country.
•
So the “or” used in this case is inclusive “or”.
Part (ii):
1. We have:
p: The school is closed if it is a holiday or a Sunday.
2. The component statements are:
q: The school is closed if it is a holiday.
r: The school is closed if it is a Sunday.
♦ q and r are connected by “or”.
3. Let us analyze the component statements:
•
On a holiday, the school is closed.
•
On a Sunday, the school is closed.
•
On a day, if Sunday and holiday coincide, then also the school is closed.
•
So the “or” used in this case is inclusive “or”.
Part (iii):
1. We have:
p: Two lines intersect at a point or are parallel.
2. The component statements are:
q: Two lines in a plane, intersect at a point.
r: Two lines in a plane, are parallel.
♦ q and r are connected by “or”.
3. Let us analyze the component statements:
•
if q is true, then r is false.
•
if r is true, then q is false.
4. Both cannot be true at the same time.
•
Both cannot be false at the same time.
•
So the “or” used in this case is exclusive “or”.
Part (iv):
1. We have:
p: Students can take French or Sanskrit as their third language.
2. The component statements are:
q: Students can take French as their third language.
r: Students can take Sanskrit as their third language.
♦ q and r are connected by “or”.
3. Let us analyze the component statements:
•
if q is true, then r is false.
•
r is true, then q is false.
•
No student can take both languages at the same time.
•
So the “or” used in this case is exclusive “or”.
Solved example 14.7
For each of the following statements, determine whether an inclusive “or” or exclusive “or” is used. Give reasons for your answer.
(i) √2 is a rational number or an irrational number.
(ii) To enter into a public library children need an identity card from the school or a letter from the school authorities.
(iii) A rectangle is a quadrilateral or a 5-sided polygon.
Solution:
Part (i):
1. We have:
p: √2 is a rational number or an irrational number.
2. The component statements are:
q: √2 is a rational number.
r: √2 is an irrational number.
♦ q and r are connected by “or”.
3. Let us analyze the component statements:
•
if q is true, then r is false.
•
if r is true, then q is false.
4. Both cannot be true at the same time.
•
Both cannot be false at the same time.
•
So the “or” used in this case is exclusive “or”.
5. In this case, r is true. So p is true.
Part (ii):
1. We have:
p: To enter into a public library children need an identity card from the school or a letter from the school authorities.
2. The component statements are:
q: To enter into a public library children need an identity card from the school.
r: To enter into a public library children need a letter from the school authorities.
♦ q and r are connected by “or”.
3. Let us analyze the component statements:
•
If identity card is available, entry is possible.
•
If letter is available, entry is possible.
•
If both identity card and letter is available, entry is possible.
•
So the “or” used in this case is inclusive “or”.
Part (iii):
1. We have:
p: A rectangle is a quadrilateral or a 5-sided polygon.
2. The component statements are:
q: A rectangle is a quadrilateral.
r: A rectangle is a 5-sided polygon.
♦ q and r are connected by “or”.
3. Let us analyze the component statements:
•
if q is true, then r is false.
•
if r is true, then q is false.
(Recall that, a quadrilateral has only four sides)
4. Both cannot be true at the same time.
•
Both cannot be false at the same time.
•
So the “or” used in this case is exclusive “or”.
5. In this case, q is true. So p is true.
Quantifiers
Some basics can be written in 4 steps:
1. Consider the set of all rectangles.
•
There will be infinite number of rectangles in that set.
2. We know that, squares are also rectangles.
•
So some of the members in the set will be squares.
3. This fact can be written as a statement:
p: There exists a rectangle whose all sides are equal.
•
This statement means that, there is at least one rectangle whose all sides are equal.
4. The phrase “There exist” is a quantifier.
•
Quantifiers are phrases which give us an idea about the “quantity of items”
•
“There exists” gives us the idea that, there is at least one item.
Let us see another quantifier. It can be explained in steps:
1. Let S be the set of all prime numbers.
•
Then we can write S in the set-builder form as:
S = {p : p is a prime number}
2. Now, square root of any prime number will be an irrational number.
•
We want to write this as a statement. We can write it as:
q: For every prime number p, √p is an irrational number.
3. The phrase “For every” is a quantifier.
•
As we mentioned just above, Quantifiers are phrases which give us an idea about the “quantity of items”
•
“For every” gives us the idea that, all items in the set are to be taken into consideration.
•
In the previous example, we had to consider only some members of the set (members which are squares).
It is important to consider the position of the quantifier. This can be explained using an example. It can be written in 5 steps:
1. Consider two sentences:
(i) For every positive number x there exists a positive number y such that y < x
(ii) There exists a positive number y such that for every positive number x, we have y < x
(In both the sentences, “number” means, real number)
2. Let us analyze the first sentence.
•
We first consider the set P of positive numbers:
P = {x : x is a +ve number}
•
The sentence says that:
We can pick any x from that set. We will be able to find a positive number y such that, y is less than x.
•
For example, (y= x/2) will be less than x.
3. Let us analyze the second sentence.
•
We first consider the set P of positive numbers:
P = {y : y is a +ve number}
•
The sentence says that:
We can pick any y from that set. That y will be less than any positive number x.
•
This is not possible. For example, y will be greater than (x= y/2).
•
So this sentence is false.
4. Both sentences appear to be the same.
•
But we see that:
♦ First sentence is true.
♦ Second sentence is false.
5. Why is that so?
•
In the first sentence, the quantifier “there exists” is at the middle.
•
In the second sentence, “there exists” is at the beginning.
•
The quantifiers must be carefully introduced in a sentence. Each quantifier must be introduced precisely at the right place. Not too early and not too late.
The link to some solved examples is given below:
In the next section, we will see implications.
No comments:
Post a Comment