CLASS 11th
Mathematical Reasoning
Mathematical Reasoning
01. Statements or Propositions Definition A statement or a proposition is an assertive (or a declarative) sentence which is either true of false but not both. A true statement is also known as a valid statement. If a statement is false, we say that it is an invalid statement. Open Statement A declarative sentence containing variable (s) is an open statement if. It becomes a statement when the variable (s) is (are) replaced by some definite value (s). Truth Set The set of all those values of the variable (s) in an open statement for which it becomes a true statement is called the truth set of the open statement. Truth Value The truth or falsity of a statement is called its truth value. Simple Statement Any statement or proposition whose truth value does not explicity depend on another statement is said to be a simple statement. Compound Statements If a statement is combination of two or more simple statement, then it is said to be a compound statement or a compound proposition. The simple statements which form a compound statement are known as its sub-statements or components or constituents. If p, q, r, .... are sub-statements of a compound statement S, then we write the compound statement as S (p, q, r, ....). The fundamental property of a compound statement is that its truth value is completely determined by the truth values of the sub-statements together with the way in which they are connected to form the compound statement. Connectives The phrases or words which connect simple statement are called logical connectives or sentenial connectives or simply connectives or logical operators. Connective and or If .... then If and only if (iff) not Remark
Symbol ∧ ∨ ⇒ or → ⇔ or ↔ ~ or ￢
Nature of the Compound statement formed by using the connective conjunction Disjunction Implication or conditional Equivalence or bio-conditional Negation
Negation is called a connective although it does not combine two or more statements. In fact, it only modifies a statement.
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Mathematical Reasoning
02. Truth Tables Truth Table A table that shows the relationship between the truth value of a compound statement S (p, q, r,...) and the truth values of its sub-statements p, q, r, ... etc, is called the truth table of statement S.
03. Basic Logical Connectives (i)
Conjunction Any two simple statements can be connected by the word “and” to form a compound statement called the conjunction of the original statements. Symbolically, if p and q are two simple statement, then p ∧ q denotes the conjunctions of p and q and is read as “p and q” For any two statement p and q, we find that p ∧ q is a statement, so it has truth value, and this truth value depends only on the truth values of p and q. Truth table for conjunction p q p ∧ q T T T T F F F T F F F F
Disjunction or Alternation Any two statements can be connected by the word “or” to form a compound statement called the disjunction of the original statements. Symbolically, if p and q are two simple statements, then p ∨ q denotes the disjunction of p and q and is read as “p and q”. p ∨ q is true only when at least one of p and q is true, otherwise it is false. Thus, we have th following truth table for p ∨ q. Truth table for disjunction p q p ∧ q T T T T F T F T T F F F Remark
The English word “or” is commonly used in two distinct ways. Sometimes it is used in the sense of “p or q or both”, i.e. at least one of the two alternatives occurs and sometimes it is used in the sense of “p or q but not both”, i.e. exactly one of the two alternatives occurs. In mathematics and also in logic, the word “or” is always inclusive. we shall usually omit the phrase “or both”, but it will always be implied.
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Mathematical Reasoning
Negation The denial of a statement p is called its negation, written as – p. Negation of any statement p is formed by writing “It is not the case that ...” or “It is false that .....” before p or, if possible by inserting in p the word “not”. Truth Table For Negation Clearly, if p is true, then ~ p is false; and if p is false, then ~ p is true. Thus, we have the following truth table for negation. p T F
~ p F T
04. Conditional And Biconditional Statements In Mathematics we come across many statements of the form “if p then q” and “p if an only if q” such statements are called conditional statements. In this section we shall discuss about such statements. (i)
Implication or Conditional Statements Two statements connected by the connective phrase ‘if ... then’ give rise to a compound statement which is known as an implication or a conditional statement. If p and q implication If p and q implication
are two statements, the compound statement ‘if p then q’ is called an or conditional statement. are two statements forming the implication ‘if p then q’ then we denote this by “p ⇒ q” or “p → q”. In the implication “p ⇒ q”, p is called the antecedent or hypothesis and q the consequent or conclusion. The truth value of an implication p ⇒ q depends on the truth values of its antecedent p and consequent q. Truth table for p ⇒ q p q p ⇒ q T T T F T T T F F F F T Remark
The truth table of p ⇒ q is same as that of ~ p ∧ q. ∴ p ⇒ q ≡ ~ p ∨ q
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Mathematical Reasoning
Biconditional Statement A statement is a biconditional statement if it is the conjunction of two conditional statements (implications) one converse to the other. Thus, if p and q are two statements, then the compound statement p if and only if q is called a biconditional statement or an equivalence and is denoted by p ⇔ q. Thus, p ⇔ q : (p ⇒ q) ∧ (q ⇒ p) Since p ⇒ q is the conjunction of p ⇒ q and q ⇒ p. So, we have the following truth table for p ⇔ q. p T T F F
q T F T F
p ⇒ q q ⇒ p T T F T T F T T
p ⇔ q = (p ⇒ q) ∧ (q ⇒ p) T F F T
05. Tautologies and Contradictions Statement Pattern A compound statement with the repetitive use of the logical connectives is called a statement pattern or a well-formed formula. Tautology A statement pattern is called a tautology, if it is always true, whatever may be the truth values of constitute statements. A tautology is called a theorem or a logically valid statement pattern. A tautology contains only T in the last column of its truth table. Contradiction A statement pattern is called a contradiction, if it is always false, whatever may the truth values of its constitute statements. In the last column of the truth table of contradiction there is always F.
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Mathematical Reasoning
JEE Main Pattern Exercise (1) 1. Consider the following propositions: p : I take medicine, q : I can sleep Then, the compound statement ~ p → ~ q means (a) If I do not take medicine, then I cannot sleep (b) If I do not take medicine, then I can sleep (c) I take medicine iff I can sleep (d) I take medicine if I can sleep 2. If the inverse of implication p → q is defined as ~ p → ~ q, then the inverse of the proposition (p ∧ ~ q) → r is (a) ~ r → ~ p ∨ q (b) ~ p ∨ q → ~ r (c) r → p ∧ ~ q (d) none of these 3. The logically equivalent proposition of p ↔ q is (a) (p ∧ q) ∨ (p ∨ q) (b) (p → q) ∧ (q → p) (c) (p → q) ∨ (q → p) (d) (p ∧ q) → (p ∨ q) 4. If p → (q ∨ r) is false, then the truth values of p, q, r are respectively (a) T, F, F (b) F, F, F (c) F, T, T (d) T, T, F 5. (p ∧ (a) (b) (c) (d)
~ q) ∧ (~ p ∨ q) is a tautology a contradiction both a tautology and a contradiction neither a tautology nor a contradiction
6. Which (a) (b) (c) (d)
of the following is always true? (p → q) ≅ (~ q → ~ p) ~ (p ∨ q) ≅ (~ p ∨ ~ q) ~ (p → q) ≅ (p ∨ ~ q) ~ (p ∧ q) ≅ (~ p ∧ ~ q)
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Mathematical Reasoning 7. The contrapositive of the statement “if 22 = 5, then I get first class” is (a) If I do not get a first class, then 22 = 5 (b) If I do not get a first class, then 22 ≠ 5 (c) If I get a first class, then 22 = 5 (d) none of these 8. Which of the following is wrong? (a) p → q is logically equivalent to ~ p ∨ q (b) If the truth values of p, q, r are T, F, T respectively, then the truth value of (p ∨ q) ∧ (q ∨ r) is T (c) ~ (p ∨ q ∨ r) ≅ ~ p ∧ ~ q ∧ ~ r (d) The truth value of p ∧ ~ (p ∨ q) is always T 9. Let S be a non-empty subset of R. Consider the following statement: P : There is a rational number x ∊ S such that x > 0. (a) Every rational number x ∊ S such that x ≤ 0 (b) x ∊ S and x ≤ 0 ⇒ is not rational (c) There is a rational number x ∊ S such that x ≤ 0 (d) There is no rational number x ∊ S such that x ≤ 0 10. Consider the following statements: P : Suman is brilliant Q : Suman is rich R : Suman is honest The negation of the statement “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as: (a) ~ P ∧ (Q ↔ ~ R) (b) ~ (Q ↔ (P ∧~ R)) (c) ~ Q ↔ (~ P ∧ R) (d) ~ (P ∧ ~ R) ↔ Q
ANSWER Q1 (a)
Q2 (b)
Q3 (b)
Q4 (a)
Q5 (b)
Q6
Q7
Q8
Q9
Q10
(a)
(b)
(d)
(a)
(b)
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