ML Aggarwal Class 11 Maths Solutions Chapter 14 Mathematical Reasoning

Introduction to Mathematical Reasoning

In ancient Greece, travelling educators called Sophists built a reputation for being able to support any position. They would confuse listeners with arguments like:

Laika is my cat.
Laika is a mother.
Therefore, Laika is my mother.

To help reveal the clever tricks employed by Sophists, Aristotle created a work called 'Refutation of the Sophists'. This work describes many faulty reasoning patterns (called fallacies), which work well for spotting weak arguments. In general, studying Logic means building strong arguments and identifying weak arguments. So, Logic is the study of how we reason.

Though Aristotle was among the first people to write about Logic, English mathematician George Boole (1815 - 1864) was the first to use mathematical methods in studying Logic. For this reason, the mathematical study of logic is sometimes referred to as Boolean logic.

14.1 Propositions (Statements)

To begin studying Geometry, you begin with basic ideas like points and lines. The study of logic is built on the foundation of a proposition.

It is important to tell the difference between three related ideas:

Utterance: Any spoken or written expression, including meaningless phrases such as "gubbledook pinaca", as well as sentences.

Sentence: Any utterance that has meaning. Sentences can include questions ("Where are you going?"), orders ("Eat your lunch!"), feelings ("Hip, hip, hurray!"), pleas ("Give me a glass of water"), hopes ("Wish you best of luck"), and propositions.

Proposition (Statement or Assertion): Any sentence that is either correct or incorrect. Examples include:

(i) Two plus two equals four.
(ii) π is a rational number.
(iii) Indira Gandhi was first woman President of India.
(iv) An elephant weighs more than a human being.
(v) It was raining in Delhi during new year's eve in 2004.

We can right away see that the first and fourth sentences are right (correct) while the second and third sentences are wrong (incorrect). There is no ambiguity. However, even though we may not instantly know if the fifth sentence above is correct or incorrect, it is clear that it will have only one value - correct or incorrect. It cannot be both correct and incorrect at the same moment. So it is also a statement.

Notice that a statement is either correct or incorrect, or in other words, either valid or not valid. It cannot be both correct and incorrect at the same time. This fact is called the law of the excluded middle.

Thus a sentence is called a mathematically acceptable statement (or proposition) if it is either correct or incorrect but not both.

Now look at these sentences:

(i) Girls are more intelligent than boys.
(ii) Sum of two integers x and y is greater than 0.
(iii) Product of two integers x and y is greater than 0.
(iv) Open the door.
(v) Where were you born?
(vi) What a lovely day!
(vii) Tomorrow is sunday.
(viii) It was raining yesterday.
(ix) He is a Mathematics teacher.
(x) Srinagar is far from here.

None of these sentences is a statement (proposition). Regarding the first sentence, some people may think that it is correct while some people may think that it is incorrect. This shows that it is an unclear sentence and hence it is not a statement.

The second sentence may be correct for some values of x and y (e.g. x = 2, y = 3) while it may be incorrect for some values (e.g. x = - 2, y = 1). So the second sentence is not a statement. The third sentence may by correct for some values of x and y (e.g. x = 3, y = 5) while it may be incorrect for some values (e.g. x = - 3, y = 5). So the third sentence is not a statement.

Sentences (iv), (v), (vi) are not statements because they are a command, a question and an exclamation in that order.

Sentences (vii) and (viii) are not statements. Sentences with words like today, yesterday, tomorrow are not treated as statements.

Sentence (ix) is not a statement, because it is not clear who "he" is. Similarly, (x) is not a statement because it is not clear what "here" means. So sentences with personal pronouns where the person is not clearly named and sentences with unspecified places ('here', 'there') and so forth are not mathematically acceptable statements.

Are the following propositions (statements) or not?

1. Ob la di ob la da.
2. Left, Right, Left!
3. Good morning to all.
4. Raise your hands!
5. Are you going to Delhi?
6. Have you ever gone to America?
7. May God bless you!
8. How tall is Bobby?
9. Bobby is 3 metres tall.
10. Sania plays tennis.
11. The sun is a star.
12. All roses are white.
13. All integers are natural numbers.
14. All squares are rectangles.
15. 5 + 3 = 10.
16. 5 + 3 = 8.
17. 9 is greater than 14.
18. There are 40 days in a month.

(Answer: Last 10 are propositions, first 8 are not)

By convention, propositions are shown using lower case letters p, q, r and so on (with or without subscripts). 'True' or 'False' (shortened to T and F in that order) is called the truth value or logical value of any statement based on whether the statement is correct or incorrect.

So the statements 'The sum of 3 and 5 is 8' and '3 + 5 ≥ 7' are correct and have truth value "True" or "T". But the statements '3 + 5 = 10' and 'Every set is finite' are incorrect and have truth value "False" or "F".

Now think about the sentence "x + 3 = 5". The truth value of this sentence cannot be found until we know the value of x. Such sentences are called "open sentences". They are not statements.

Think about the sentences "x + 0 = x" and "(x - 1) (x + 1) = x² - 1". No matter what value x takes, they always have truth value "True". So, algebraic identities are statements (propositions).

Exercise 14.1

Very short answer type questions (1 to 3):

Question 1. Identify which of the following are statements (propositions):
(i) Krishna is black.
(ii) How black is Krishna?
(iii) Listen to me!
(iv) A triangle has four sides.
(v) The moon revolves around the earth.
(vi) Prime factors of 6 are 2 and 3.
(vii) Do you know the prime factors of 6?
(viii) Do your homework.
(ix) x² + 5x + 6 = 0.
(x) x² + 5x + 6 = (x + 2)(x + 3).
(xi) 2 + 2 = 4.
(xii) 2 + 2 = 5.
(xiii) 2 + 5 < 11.
(xiv) The earth is a star.
Answer: The statements are (i), (iv), (v), (vi), (x), (xi), (xiii), and (xiv). The non-statements are (ii) - a question, (iii) - an order, (vii) - a question, (viii) - a command, and (ix) - an open sentence that depends on the value of x.
In simple words: A statement must be a sentence that is either right or wrong, with no doubt. Questions, orders, and sentences where we don't know what x is are not statements.

Exam Tip: Look for questions, commands, exclamations, and open sentences (with variables like x, y) - none of these are statements. Every true statement (proposition) can be marked as either true or false with total certainty.

Question 2. Which of the following sentences are statements? Give reasons for your answer.
(i) Mathematics is fun.
(ii) The sun is a star.
(iii) There is no rain without clouds.
(iv) How far is Mumbai from here?
(v) Mathematics is difficult.
(vi) The square of an even integer is an even integer.
(vii) Answer this question.
(viii) Today is a windy day.
(ix) The sides of a rhombus have equal lengths.
(x) There are 35 days in a month.
(xi) All real numbers are complex numbers.
(xii) All complex numbers are real numbers.
(xiii) There are twelve days in a week.
(xiv) The square of an odd integer is an even integer.
(xv) Tomorrow is holiday.
(xvi) x - 12 = 5.
(xvii) She is a mathematics graduate.
(xviii) Every square is a rectangle.
(xix) sin² x + cos² x = 0.
Answer: The statements are (ii), (iii), (vi), (ix), (x), (xi), (xii), (xiii), and (xviii). These can be marked as definitely right or definitely wrong. The non-statements are: (i) "Mathematics is fun" - depends on personal opinion; (iv) - a question; (v) "Mathematics is difficult" - depends on personal view; (vii) - a command; (viii) "Today is a windy day" - uses "today" which changes daily; (xiv) - wrong but still a valid statement that can be marked false; (xv) - uses "tomorrow" which changes daily; (xvi) - an open sentence depending on x; (xvii) - unclear who "she" refers to; (xix) - can be checked and is a statement (it is false).
In simple words: Statements must be clear sentences that are always either right or wrong, no matter when or where. Sentences with "I like it", "today", "tomorrow", unclear pronouns, questions, or commands are not statements.

Exam Tip: Watch for hidden non-statements: personal opinions (fun, difficult), time-dependent words (today, tomorrow, yesterday), unclear reference words (he, she, here), and questions/commands. These trap many students.

Question 3. State the truth values of the following:
(i) There are only finite number of rational numbers.
(ii) The quadratic equation ax² + bx + c = 0, a ≠ 0, a, b, c ∈ ℝ, has always two real roots.
(iii) Sky is red.
(iv) New Delhi is the capital of India.
(v) Every rectangle is a square.
(vi) Zero is a complex number.
(vii) 2 + 2 = 4.
(viii) 2 + 2 = 5.
(ix) x - 0 = x.
(x) The equation x² - 1 = 0 has two roots, +1 and -1.
(xi) Pigs fly.
(xii) x³ + 1 = (x + 1)(x² - x + 1).
Answer: (i) False - (ii) False - (iii) False - (iv) True - (v) False - (vi) True - (vii) True - (viii) False - (ix) True - (x) True - (xi) False - (xii) True
In simple words: Go through each sentence one at a time. If it is correct, write T (True). If it is incorrect, write F (False). This is the truth value.

Exam Tip: Algebraic identities like (xii) always have truth value True. Be careful with statements using variables - (ix) is true for ALL numbers, so it is a true statement (an identity).

Question 4. Fill in the blanks:
(i) Logic is the study of ______
(ii) Axiomatic (mathematical) approach to the study of logic was first used by ______
(iii) Imperative, exclamatory and interrogative sentences are not ______
(iv) The law of the excluded middle says that ______
(v) An open sentence is a statement. True or False? ______
(vi) Algebraic identities are statements. True or False? ______
Answer: (i) reasoning - (ii) George Boole - (iii) statements (propositions) - (iv) a statement cannot be both true and false at the same time - (v) False - (vi) True
In simple words: Remember: Logic teaches us to reason well. George Boole brought math into logic. Orders, questions, and exclamations are not statements. A statement is either right or wrong, never both. Open sentences (with x or y) are not statements, but formulas that work for all numbers are.

Exam Tip: These are definition-type fill-in-the-blank questions. George Boole is a key historical name to remember. Open sentences are a common trap - students often think "x + 3 = 5" is a statement, but it is not.

Question 5. Give five examples of sentences which are not statements. Give reason for the answers.
Answer: Examples include: (1) "Is it raining?" - This is a question, and questions cannot be marked true or false. (2) "Close the window!" - This is a command or order, and orders are not statements. (3) "What a beautiful sunset!" - This is an exclamation expressing feeling, not a fact that can be checked as right or wrong. (4) "Today is Monday." - This sentence uses the word "today" which changes depending on which day you say it, so it is not fixed. (5) "He is very tall." - We do not know who "he" is, so the sentence is unclear and cannot be marked definitively true or false.
In simple words: A sentence is not a statement if it is a question, command, or feeling; if it uses changing words like "today"; or if it has unclear words like "he" or "here".

Exam Tip: For this question, give a different reason each time to show you understand what makes a statement. The five main reasons are: question type, command type, exclamation type, time-dependent words, and unclear reference words.

14.2 Negation of a Statement

The refusal or opposite of a statement is called the negation of the statement.

Given the statement p: "Raja is happy", look at these statements:

(i) Raja is not happy.
(ii) Raja is unhappy.
(iii) Raja is sad.
(iv) It is false that Raja is happy.
(v) It is not true that Raja is happy.
(vi) It is not the case that Raja is happy.

Each of these statements has the opposite meaning from the original statement.

This is known as negation of proposition p, and is indicated by ~ p, p′ or p with a line over it. Since the symbol ~ applies to just one statement, it is sometimes known as a modifier rather than a connective.

So, negation of "Sky is purple" is "Sky is not purple"; negation of "3 + 3 = 5" is "3 + 3 ≠ 5" or "It is false that 3 + 3 = 5"; negation of "x ∈ A" is "x ∉ A", and so on.

Remark

The opposite of statements that use words like "for all", "there is at least one", "some" or "for every" can be tricky. For example:

(i) The negation of "All mathematicians are men" is not "All mathematicians are not men". In fact, the negation of the given statement is "Not all mathematicians are men" or "There is at least one mathematician who is not a man" or "It is false that all mathematicians are men".

(ii) The negation of "There is at least one dog which does not bite" is "There does not exist a dog which does not bite" or "All dogs bite".

Now look at the truth/false value of these statements:

Now look at the opposites of these statements (propositions) and their truth/false value:

See what took place. Negation changes a correct statement into a false statement, and a false statement into a correct statement. In other words, if p is correct, then ~ p is false; if p is false, then ~ p is correct. This can be shown in a table as:

Such tables are called truth tables. Note that the logical value 'T' stands for 'True' and the logical value 'F' stands for 'False'.

Can you write the opposite of these sentences along with their logical value (T or F)?

(i) 1 + 1 = 2
(ii) 1 + 1 = 0
(iii) 2 ∈ ℕ
(iv) New Delhi is the capital of India
(v) Pigs fly
(vi) All natural numbers are integers

Illustrative Examples

Question. Example 1. Write the negation of the following statements. Also check whether the resulting statements are true or false.
(i) 7 is rational.
(ii) Sum of 3 and 4 is 9.
(iii) Every natural number is greater than 0.
(iv) Australia is a continent.
Answer:
(i) The opposite of the given statement is: It is false that 7 is rational. It can also be stated as: 7 is not rational or 7 is irrational. We know this is correct. (Notice that the original statement was false).
(ii) The opposite of the given statement is: It is false that the sum of 3 and 4 is 9. It can also be stated as: The sum of 3 and 4 is not 9. We know this is correct. (Notice that the original statement was false).
(iii) The opposite of the given statement is: It is false that every natural number is greater than 0. It can also be stated as: There is at least one natural number which is not greater than 0. We know this is false. (Notice that the original statement was correct).
(iv) The opposite of the given statement is: It is false that Australia is a continent. It can also be stated as: Australia is not a continent. We know this is false. (Notice that the original statement was correct).
In simple words: To negate a statement, put "not" into it or say "It is false that...". If the original was right, the opposite is wrong. If the original was wrong, the opposite is right.

Exam Tip: When negating a statement with "all" or "every", change it to "not all" or "there exists at least one...". Do not just add "not" to the whole thing - that is a common error.

Question. Example 2. Write the negations of the following statements:
(i) Everyone in France speaks French.
(ii) Both the diagonals of a rectangle have the same length.
(iii) There does not exist a quadrilateral which has all sides equal.
Answer:
(i) It is false that everyone in France speaks french. Another way: There is at least one person in France who does not speak French.
(ii) It is false that both the diagonals of a rectangle have the same length. Another way: There is at least one rectangle whose both diagonals do not have the same length.
(iii) It is not the case that there does not exist a quadrilateral which has all sides equal. Another way: There is at least one quadrilateral which has all sides equal.
In simple words: When you negate "Everyone does X", it becomes "Not everyone does X" or "At least one person does not do X". When you negate "There does not exist", it becomes "There exists".

Exam Tip: Be careful with double negatives. "There does not exist..." negated becomes "There exists...", not "There does not exist... not...". Work slowly through the logic.

Exercise 14.2

Very short answer type questions (1 to 4):

Question 1. Write the negation of the following statements:
(i) The number 17 is prime.
(ii) 5 is a rational number.
(iii) 2 is not a complex number.
(iv) 2 + 7 = 6.
(v) The number 2 is greater than 7.
(vi) Cow has four legs.
(vii) A leap year has 366 days.
(viii) Every natural number is an integer.
(ix) Every real number is an irrational number.
(x) All triangles are not equilateral.
(xi) All similar triangles are congruent.
(xii) Area of every circle is the same as the perimeter of the circle.
Answer: (i) The number 17 is not prime - (ii) 5 is not a rational number - (iii) 2 is a complex number - (iv) 2 + 7 ≠ 6 - (v) The number 2 is not greater than 7 - (vi) Cow does not have four legs - (vii) A leap year does not have 366 days - (viii) It is false that every natural number is an integer. Alternatively: There is at least one natural number which is not an integer - (ix) There is at least one real number which is not an irrational number - (x) It is false that all triangles are not equilateral - (xi) There exist similar triangles which are not congruent - (xii) There exists a circle whose area is not the same as the perimeter of the circle
In simple words: To negate most statements, add "not" to make it the opposite. For "every" or "all" statements, use "There is at least one... not" instead.

Exam Tip: Statement (x) is tricky - "All triangles are not equilateral" means "No triangle is equilateral", so its negation is the opposite: "At least one triangle is equilateral".

Question 2. Are the following pairs of statements negations of each other?
(i) The number x is not a rational number. The number x is not an irrational number.
(ii) The number x is not a rational number. The number x is an irrational number.
Answer: (i) No - these are not negations of each other. x can be a complex number, for example, and be neither a rational number nor an irrational number (if x = 2i). - (ii) Yes - these are negations of each other. Every number is either rational or irrational, so saying "x is not rational" is the same as saying "x is irrational"
In simple words: Two statements are negations if one is the exact opposite of the other. Ask yourself: if the first is right, must the second be wrong? And if the first is wrong, must the second be right?

Exam Tip: Remember that "not rational" and "irrational" mean the same thing for real numbers, since real numbers are either rational or irrational. But for complex numbers, they do not mean the same thing.

Question 3. Write the negation of "All men are mortal"
(i) Using "There exists atleast..."
(ii) Without using "There exists..."
Answer: (i) There exists at least one man who is not mortal - (ii) Not all men are mortal. Alternatively: It is false that all men are mortal
In simple words: Negating "All X are Y" can be done two ways. Way 1: "There is at least one X that is not Y". Way 2: "Not all X are Y".

Exam Tip: These two forms are equivalent. Pick whichever your teacher or textbook uses, but know both ways to write the same idea.

Question 4. Write down the negation of the statement "All the sides of an equilateral triangle are of the same length".
Answer: There is at least one equilateral triangle all of whose sides are not of the same length.
In simple words: Instead of "All sides are the same", say "At least one side is different".

Exam Tip: This is a definition statement about equilateral triangles. The original statement is always true (by definition), so its negation is always false.

14.3 Compound Statements

Simple statement: A statement that cannot be divided into two or more statements is called a simple statement. For example:

(i) Roses are red
(ii) New Delhi is the capital of India
(iii) Every set is a finite set

are all simple statements.

Compound statement: A statement that is made by joining two or more simple statements is called a compound statement.

Each statement within a compound statement is called a component statement.

Look at this statement:

p: Sam is very smart or he is very lucky.

This statement is really made of two (component) statements joined by "or":

p: Sam is very smart.
r: Sam is very lucky.

Now look at this statement:

p: Delhi is in India and Islamabad is in Pakistan.

This statement is made of two simple statements joined by "and":

p: Delhi is in India.
r: Islamabad is in Pakistan.

Similarly, the sentence "Navin likes tennis and cricket" is made of two simple statements "Navin likes tennis" and "Navin likes cricket" joined by "and".

Question. Example 1. Find the component statements of the following compound statements:
(i) Zero is a positive number or a negative number.
(ii) There is something wrong with the bulb or the wiring.
(iii) Sun is bigger than earth and earth is bigger than moon.
(iv) All rational numbers are real and all real numbers are complex.
Answer:
(i) The component statements are: p: Zero is a positive number. q: Zero is a negative number. The joining word is "or".
(ii) The component statements are: p: There is something wrong with the bulb. q: There is something wrong with the wiring. The joining word is "or".
(iii) The component statements are: p: Sun is bigger than earth. q: Earth is bigger than moon. The joining word is "and".
(iv) The component statements are: p: All rational numbers are real. q: All real numbers are complex. The joining word is "and".
In simple words: Break the compound statement into its parts. Look for the words "and" or "or" to find where one part ends and the next begins. Write each part as its own simple statement.

Exam Tip: Always identify the joining word first - this tells you what type of compound statement it is. Then separate the statement into its two (or more) parts.

Question. Example 2. Find the component statements of the following and check whether they are true or not:
(i) All prime numbers are either even or odd.
(ii) A square is a quadrilateral and its four sides are equal.
(iii) Chandigarh is the capital of Haryana and U.P.
(iv) 100 is divisible by 3, 5 and 11.
Answer:
(i) The component statements are: p: All prime numbers are even. q: All prime numbers are odd. Both are false. The joining word is "or".
(ii) The component statements are: p: A square is a quadrilateral. q: A square has all its four sides equal. Both are correct. The joining word is "and".
(iii) The component statements are: p: Chandigarh is the capital of Haryana. q: Chandigarh is the capital of U.P. The first is correct and the second is false. The joining word is "and".
(iv) The component statements are: p: 100 is divisible by 3. q: 100 is divisible by 5. r: 100 is divisible by 11. The second is correct while the first and third are false. The joining word is "and".
In simple words: After you break it up, check if each part is right or wrong. Write down the truth value for each one.

Exam Tip: When a compound statement uses "and" with three parts (like example iv), all three parts must be true for the whole thing to be true. If even one is false, the whole thing is false.

Exercise 14.3

Very short answer type questions:

Question 1. Find the component statements of the following compound statements. Clearly mention the connecting word.
(i) Jack and Jill went up the hill.
(ii) Chennai is in India and is the capital of Tamil Nadu.
(iii) 7 is a rational number or an irrational number.
(iv) Number 7 is odd and prime.
(v) A rectangle is a quadrilateral or a 5-sided polygon.
Answer:
(i) The component statements are: p: Jack went up the hill. q: Jill went up the hill. Joining word: "and".
(ii) The component statements are: p: Chennai is in India. q: Chennai is the capital of Tamil Nadu. Joining word: "and".
(iii) The component statements are: p: 7 is a rational number. q: 7 is an irrational number. Joining word: "or".
(iv) The component statements are: p: Number 7 is odd. q: Number 7 is prime. Joining word: "and".
(v) The component statements are: p: A rectangle is a quadrilateral. q: A rectangle is a 5-sided polygon. Joining word: "or".
In simple words: Split the compound statement. Look for "and" or "or". Rewrite each part as a full simple statement. Name the joining word.

Exam Tip: Be careful with statements like (i) - "Jack and Jill went up the hill" can be split into two statements even though there is only one verb at the end. This is a common pattern.

Question 2. Find the component statements of the following compound statements and check whether they are true or false.
(i) All integers are positive or negative.
(ii) 2 is a rational number or an irrational number.
(iii) A student who has passed Mathematics or Computer Science can go for MCA.
(iv) 36 is a multiple of 2, 6 and 8.
Answer:
(i) The component statements are: p: All integers are positive. q: All integers are negative. Both are false (since zero is an integer but is neither positive nor negative). Joining word: "or".
(ii) The component statements are: p: 2 is a rational number. q: 2 is an irrational number. The first is correct and the second is false. Joining word: "or".
(iii) The component statements are: p: A student who has passed Mathematics can go for MCA. q: A student who has passed Computer Science can go for MCA. Both can be accepted as correct or depends on university rules. Joining word: "or".
(iv) The component statements are: p: 36 is a multiple of 2. q: 36 is a multiple of 6. r: 36 is a multiple of 8. The first two are correct but the third is false (36 ÷ 8 = 4.5, not a whole number). Joining word: "and".
In simple words: Break the statement apart, then check each part using your math knowledge. Write T (True) or F (False) for each one.

Exam Tip: In example (i), remember that zero is an integer but is neither positive nor negative, so "All integers are positive or negative" is false.

14.4 Logical Connectives and Quantifiers

In Arithmetic, you know operations on numbers, like + (addition), × (multiplication), ÷ (division) and so on. In sets, you have used operations like complement (Ā, A^c and so on), union (A ∪ B), intersection (A ∩ B) and so on. In mathematical reasoning, simple statements can be joined by using connectives like 'And', 'Or', 'Implies' and so on.

14.4.1 The connective 'And' - Conjunction (p ∧ q)

When two simple statements are combined using the word "and" to form a compound statement, the result is called the conjunction of the original statements. In symbols, the conjunction of the two statements p and q is shown by p ∧ q. The items p and q of p ∧ q are called its conjuncts.

If p is 'Krishna is rich' and q is 'Sudama is poor', then p ∧ q is 'Krishna is rich and Sudama is poor'. Notice that the conjuncts can be switched around without changing the meaning. So 'Sudama is poor and Krishna is rich' means the same as the previous sentence. In other words, q ∧ p has the same meaning as p ∧ q. In daily speech, we use conjunctions often, but sometimes in hidden ways. Some typical hidden forms are:

Also, sentences such as 'Mango and Banana are fruits' can be reworded as 'Mango is a fruit and Banana is a fruit'.

Note that the word 'And' in a statement does not always mean it can be split into two smaller statements. Look at this sentence:

Oil and water do not mix well.

Here the word 'And' is not a joining word, and the given statement cannot be split into two smaller statements - it is a simple statement.

Logical value of conjunction

p ∧ q is correct only if both the parts p and q are correct, or else it is false. Look at these four statements:

(i) The sun rises in the east and sets in the west.
(ii) The sun rises in the north and sets in the south.
(iii) 5 > 3 and 7 < 5.
(iv) 2 + 2 = 5 and London is in England.

Only the statement (i) is correct as both of its parts are correct. The other three statements are false as at least one of the smaller statements is false.

Truth table of a conjunction is:

So, a compound statement with joining word(s) 'AND' is correct only if all of its component statements are correct; otherwise, it is false.

Question. Example. Write the component statements of the following compound statements and check whether the compound statement is true or false.
(i) A point occupies a position and its location can be determined.
(ii) 2 + 2 = 5, whereas 3 + 3 = 6.
(iii) 42 is divisible by 5, 6 and 7.
(iv) 42 is a multiple of 2, 3 and 7.
Answer:
(i) The component statements of the given statement are: p: A point occupies a position. q: A point's location can be determined. The joining word is "and". Since both component statements are correct, the given compound statement is correct.
(ii) The given conjunction is in hidden form - the word "whereas" can be replaced by "and". So the component statements are: p: 2 + 2 = 5 q: 3 + 3 = 6. Since the first statement is false, the given compound statement is false.
(iii) The given compound statement is made of three component statements joined by 'and': p: 42 is divisible by 5 q: 42 is divisible by 6 r: 42 is divisible by 7. The second and third component statements are correct but the first statement is false. So the given compound statement is false.
(iv) The given compound statement is made of three component statements joined by 'and' p: 42 is a multiple of 2 q: 42 is a multiple of 3 r: 42 is a multiple of 7. All three statements are correct. So the given compound statement is correct.
In simple words: To find if an "and" statement is right, check each part. If all parts are right, the whole thing is right. If even one part is wrong, the whole thing is wrong.

Exam Tip: Watch for hidden joining words like "whereas", "although", "but", and "besides" - these all act like "and" in logic. Also, remember that "42 is divisible by 5" is false, so the whole statement in (iii) is false.

14.4.2 The connective "or" - Disjunction (p ∨ q)

When two simple statements are combined by the word "or" to form a compound statement, the result is called the disjunction of the original statements. In symbols, the disjunction of the two statements p and q is shown by p ∨ q. The items p and q of p ∨ q are called its conjuncts.

If p is "The sun shines" and q is '6 is prime', then p ∨ q is 'The sun shines or 6 is prime'.

Remark

The way the word "or" is used in the English language is not clear-cut.