JEE Mathematics Mathematical Reasoning Notes Set 01

Statements and Truth Values

A statement (proposition) is a sentence that is either true or false but not both. A true statement is called valid and a false statement is called invalid.
Commands, requests, questions, exclamations and wishes are not statements. Sentences whose truth depends on an unspecified quantity or on a changing time or place (for example 'x + y is greater than 0', 'tomorrow is Monday', 'it is cold here') are also not statements. Scientifically established facts are accepted as true.
The truth or falsity of a statement is its truth value. For n statements a truth table has 2ⁿ rows (2 rows for p; 4 rows for p, q; 8 rows for p, q, r).

Example 1. Which of the following sentences are statements? (i) Three plus four equals six. (ii) All prime numbers are odd. (iii) The sum of x and y is greater than 0. (iv) Is every set finite?
Answer:
(i) is a false sentence and (ii) is a false sentence, yet each has a definite truth value, so both are statements.
(iii) cannot be judged true or false until x and y are known, so it is not a statement.
(iv) is a question, so it is not a statement.

Negation

The negation of a statement p, written ~p and read 'not p', denies p. It can be formed by writing 'it is not the case that p' or by inserting 'not'. ~p is false when p is true and true when p is false.
The negation of a statement with the word 'all' (or 'every') is a statement with 'there exists' (or 'at least one') together with the denied property, and the negation of 'there exists' is 'for all' with the denied property.

Example 2. Write the negation of (i) 'All cats scratch' (ii) '√5 is a rational number'.
Answer:
(i) At least one cat does not scratch (some cats do not scratch).
(ii) √5 is an irrational number.

Compound Statements and the Connectives And, Or

A statement made from two or more statements joined by words such as 'and', 'or', 'if ... then', 'if and only if' is a compound statement; the parts are its component statements.
Conjunction p ∧ q (p and q) is true only when both p and q are true. Disjunction p ∨ q (p or q) is false only when both p and q are false.
Types of 'or': in an exclusive 'or' the happening of one part rules out the other (I will go to Delhi either by bus or by train); in an inclusive 'or' both parts may happen together (you may take physical education or computers as an optional subject).

Example 3. Find the truth value of '2 divides 4 and 3 + 7 = 8', and write the component statements of 'All living things have two legs and two eyes'.
Answer:
'2 divides 4' is true and '3 + 7 = 8' is false, so the conjunction is false.
The components are: all living things have two legs; all living things have two eyes.

Implications: Conditional, Converse, Contrapositive, Biconditional

The conditional 'if p then q', written p ⇒ q, has hypothesis (antecedent) p and conclusion (consequent) q. It is false only when p is true and q is false.
The following all say the same as p ⇒ q: p implies q; p only if q; q if p; p is sufficient for q; q is necessary for p; whenever p is true q is true; ~q ⇒ ~p.
Related statements: converse q ⇒ p; inverse ~p ⇒ ~q; contrapositive ~q ⇒ ~p. A statement is equivalent to its contrapositive (and the converse is equivalent to the inverse), but a statement is not equivalent to its converse.
The biconditional 'p if and only if q', written p ⇔ q, means (p ⇒ q) ∧ (q ⇒ p). It is true exactly when p and q have the same truth value, and it can be read as 'p is necessary and sufficient for q'.

Example 4. Write the converse of 'If two lines are parallel then they do not intersect in the same plane', and the contrapositive of 'If Mohan is a poet, then he is poor'.
Answer:
Converse: if two lines do not intersect in the same plane then they are parallel.
With p : Mohan is a poet and q : Mohan is poor, the contrapositive ~q ⇒ ~p is: if Mohan is not poor, then he is not a poet.

Tautology, Contradiction and Algebra of Statements

A compound statement that is true for every assignment of truth values to its components is a tautology; one that is false for every assignment is a contradiction (fallacy). The negation of a tautology is a contradiction and conversely.
Two statements are logically equivalent when they have the same truth table.
Laws: idempotent p ∨ p = p, p ∧ p = p; commutative and associative laws for ∧ and ∨; distributive p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) and p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r); identity p ∨ F = p, p ∧ T = p; complement p ∨ ~p = T, p ∧ ~p = F; double negation ~(~p) = p; absorption p ∨ (p ∧ q) = p and p ∧ (p ∨ q) = p.
Useful forms: p ⇒ q = ~p ∨ q; p ⇔ q = (~p ∨ q) ∧ (p ∨ ~q).

Example 5. Show that p ⇒ (p ∨ q) is a tautology.
Answer:
Build the truth table: for the rows (p, q) = TT, TF, FT, FF the value of p ∨ q is T, T, T, F.
The conditional p ⇒ (p ∨ q) then has values T, T, T, T, so it is true in every row: a tautology.
Reason in words: whenever p is true, p ∨ q is true as well, so the conditional can never be false.

Example 6. Using the laws of algebra of statements, show that (p ∨ q) ∧ ~p ≡ ~p ∧ q.
Answer:
(p ∨ q) ∧ ~p = (p ∧ ~p) ∨ (q ∧ ~p) by the distributive law.
p ∧ ~p = F, and F ∨ (q ∧ ~p) = q ∧ ~p = ~p ∧ q.

Negation of Compound Statements

De Morgan's laws: ~(p ∧ q) = ~p ∨ ~q and ~(p ∨ q) = ~p ∧ ~q.
Negation of a conditional: ~(p ⇒ q) = p ∧ ~q (the hypothesis holds and the conclusion fails).
Negation of a biconditional: ~(p ⇔ q) = (p ∧ ~q) ∨ (~p ∧ q) = p ⇔ ~q = ~p ⇔ q.
For longer expressions, first replace ⇒ and ⇔ by ∧, ∨, ~ and then apply De Morgan's laws repeatedly.

Example 7. Write the negation of (i) 'All the students completed their homework and the teacher is present' (ii) 'If my car is not in the workshop, then I can go to college' (iii) 'ABC is an equilateral triangle if and only if it is equiangular'.
Answer:
(i) ~(p ∧ q) = ~p ∨ ~q: some students did not complete their homework or the teacher is not present.
(ii) With p : my car is not in the workshop and q : I can go to college, ~(p ⇒ q) = p ∧ ~q: my car is not in the workshop and I cannot go to college.
(iii) ~(p ⇔ q) = (p ∧ ~q) ∨ (~p ∧ q): either ABC is equilateral and not equiangular, or ABC is not equilateral and it is equiangular.

Example 8. Find the negation of p ∧ ~q.
Answer:
~(p ∧ ~q) = ~p ∨ ~(~q) = ~p ∨ q.

Quantifiers

The quantifier 'there exists' (∃) says that at least one member of the given set has the property; 'for all' (∀) says that every member has it.
Negation rules: ~(for all x, P(x)) = there exists x such that ~P(x); ~(there exists x such that P(x)) = for all x, ~P(x).
Example: the negation of 'there exists x ∈ Z such that x² + x = 0' is 'for all x ∈ Z, x² + x ≠ 0'.