Introduction and Preliminaries, Set Theory PDF

# INTRODUCTION AND PRELIMINARIES, SET THEORY ## 1. INTRODUCTION The kind of logic which is used in mathematics is called deductive logic. Mathematical arguments must be strictly deductive in nature. In other words, the truth of the statement to be proved must be established assuming the truth of s...

# INTRODUCTION AND PRELIMINARIES, SET THEORY ## 1. INTRODUCTION The kind of logic which is used in mathematics is called deductive logic. Mathematical arguments must be strictly deductive in nature. In other words, the truth of the statement to be proved must be established assuming the truth of some other statements. For example, in geometry we deduce the statement that the sum of the three angles of a triangle is 180 degrees from the statement that an external angle of a triangle is equal to the sum of the other (i.e., opposite) two angles of the triangle. The kind of logic which we shall use here is bi-valued i.e., every statement will have only two possibilities, either ‘True’ or ‘False’ but not both. ## 1.2. STATEMENT OR PROPOSITION An statement is a declarative sentence which is true or false, but not both; or in other words an statement is a declarative sentence which has a definite truth value. Example. Following are statements (or propositions) : | Declarative sentence | Truth value | |---|---| | {*:*² = 36}={6, -6} | True | | 3>9 | False | | The capital of Madhya Pradesh is Bhopal | True | | Blood is red | True | | 5+4=10 | False | Following are not statements : (i) How are you? (ii) Please go from here. (iii) May God help you. ## 1.3. STATEMENT LETTERS OR SENTENCE VARIABLES We know that symbols have great importance in Mathematics and therefore symbols will be used to represent statements. **Definition.** The symbols, which are used to represent statements, are called statement letters or statement variables. Statement letters by thements usually the letters P, Q, R, ..., p, q, r, ... etc. are used. For example: (i) p^q: Bhopal is the capital of Madhya Pradesh' is denoted by the letter p. (ii) pvq: is written as follows: al is the capital of Madhya Pradesh. ## 1.4. USE OF BRACKETS The use of brackets in statements (or propositions) is very important. The meaning (or explanation) of statements is included in brackets. For example: The statements (p^q)r and p^(q⇒r) have completely different meaning. Therefore, following rules are followed for brackets in statements: Rule 1. If connective 'Not (i.e., ~)' is repeated, then bracket is not required. For example-p has the meaning as (~(~p)). Rule 2. If in a statement, the connectives of the same rank appear, then brackets are applied from the left. For examples p^q^r^s^t= [{(p^q)^r^s] ^t. Rule 3. If the connectives of different ranks appear, then first of all the bracket of that connective is removed which is of lower rank. For example: pv (gr) = pv (q→r). Here the rank (= 4) of '→' is greater than the rank (= 3) of 'v'. Therefore, this bracket cannot be removed. ## 1.5. KINDS OF SENTENCES Usually the sentences are of two kinds: (i) Simple sentence or Atomic sentence (ii) Compound sentence or Molecular Sentence (i) Simple sentence. A simple sentence or atomic sentence (ii) Compound sentence. A compound sentence A compound sentence is named on the basis of the leading connective used in it. For example : | Connective | Compound Sentence | |---|---| | ^ | Conjunctive sentence | | v | Disjunctive sentence | | → or ⇒ | Conditional sentence | | ↔ or ≡ | Bi-conditional sentence | | ~ | Negative or denial sentence | (a) In conditional sentence p⇒ q, p is called 'antecedent' or 'hypothesis' or 'premise' and q is called 'consequent'. In language, it is read as : (i) p means q (ii) if p then q, (iii) p then q, (iv) when p then q, (v) q if p, (vi) q when p, (vii) p only if q, (viii) necessary condition for p is q, (ix) sufficient condition for q is p. (b) Bi-conditional sentence pq is read as: (i) if p then q and if q then p, (ii) the necessary and sufficient condition for p is q, (iii) p if and only if q. (c) The sentences used to form a compound sentence are called its components. For example: The components of p^q are p and q. Remark. The components may either be simple sentences or compound sentences. ## 1.6. TRUTH VALUE OF A STATEMENT According to the definition of statement, a statement has a definite truth value which is either ‘True or False’. Consider the following examples: (i) 15 is an odd number. (ii) 9 is a prime number. Clearly (i) is true, therefore (i) has a definite truth value ‘True’ and it is denoted by the first letter ‘T’ of True. Similarly, (ii) is a statement whose truth value is ‘False’ and it is denoted by the first letter ‘F’ of False. ## 1.7. STATEMENT PATTERN OR STATEMENT FORM OR PROPOSITION FORM Statement pattern or statement form is an expression which is obtained by combining statement letters by the use of a finite number of logical connectives. For example: If p and q be statement letters, then following are examples of statement pattern: (i) p^q, (ii) pvq, (iii) pq, (iv) pvq-p, (v) pvqqvp. The statement pattern may be simple or composite, according as it is composed of one or more logical connectives. Thus statement patterns given by (i), (ii) and (iii) are simple while those given by (iv) and (v) are composite. ## 1.8. TRUTH VALUE FUNCTION We know that every statement has a definite and unique truth value which is either True or False. Hence every statement pattern defines a truth value function of its components whose range set is {T, F}. ## 1.9. PRINCIPAL CONNECTIVE We know that a compound sentence is formed by the use of one or more than one logical connectives. **Definition**. The logical connective, which is used in the end in the composition of compound sentence, is called a principal connective. For example: (i) The principal connective in (p^p)⇒r is ‘⇒’. (ii) The principal connective in (rp) v (pq) is ‘v’. ## 1.10. OPEN STATEMENT **Definition.** A sentence, which contains one or more variables such that when certain values are substituted for variables, becomes a statement, is called an open statement. For example: Consider x + 3 = 9. If we put 6 in place of *x*, then this sentence becomes a true statement. Such a sentence is called an open statement. ### ILLUSTRATIVE EXAMPLES Example 1. Which of the following statements are true and which are false : (i) 16+9=42+32. (ii) 9×7=21 x 3. (iii) {2,3} {2,4,6). (iv) 5∈ {1,3, 5). Solution. (i) True, since 25 = 25. (ii) True, since 63 = 63. (iii) Flase, since 3 & (2, 4, 6). (iv) True, since 5 is an element of the set {1, 3, 5). Example 2. What type of sentence is 5+x=9? For what value of *x* it will become a true statement? Solution. The given sentence 5+x=9 is an open statement. It becomes a true statement for *x* = 4. Example 3. Are the following propositions? (i) Some roses are black. (ii) All roses are white. (iii) May you live long. (iv) The girls are beautiful. (v) Go to home. (vi) What is your good name ? (vii) Bravo! you have done a good work. (viii) Children, come to me. (ix) Is 7 a prime number? (x) 10 is a prime number. (xi) Solution. (i) Yes, since it is a declarative sentence, whose truth value is True (ii) Yes, since it is a declarative sentence, whose truth value is False. (iii) No, since it is not a declarative sentence. (iv), (v), (vi), (vii), (viii), (ix) and (xi) are not propositions since these all are not declarative sentences. (x) Yes, since it is a declarative sentence, whose truth value is False. Example 4. If p= he is poor, q=he is laborious, then write down the following statements in symbols (i.e., in the language of logic): (ⅰ) He is poor and laborious. (ii) He is poor but is not laborious. (iii) It is false that he is poor or laborious. (iv) It is false that he is not poor or is laborious. (v) Neither he is poor nor he is laborious. (vi) He is poor or is not poor and is laborious. (vii) It is not true that he is not poor or is not laborious. Solution. In terms of statement letters p, q and logical connectives, the solutions are as follows: (i) p^q. (ii) p^-q. (iii) - (pvq). (iv) -(-pvq). (v) -p^-9. (vi) pv (-p^q). (vii)~(~pv-q). Example 5. Write the following sentences in symbols : (i) When Sheela will come then I shall go to college. (ii) Until Sheela will not come I shall not go to college. Solution. Let p = Sheela will come, q = I shall go to college. : (i) pq. (ii) ~p-9. Example 6. Write the following in symbols : (i) What so ever I say, he refuses from that. (ii) I shall reach the station in time, otherwise I shall miss the train. (iii) I shall go to Delhi, but I shall not see zoo. (iv) In the position when he is ill, I shall wire. (v) Whenever I take leave, I fall ill. (vi) Until I shall not be called till then I shall remain here. (vii) Ram will go to work or will remain in the house and he will white-wash the house. (viii) Not only men, but also women and children were killed. (ix) If he will do labour, he will success. If he will success, he will get pleasure. Thus by doing labour one gets pleasure. (x) If teams do not arrive or the whether is bad, then there will be no match. (xi) The necessary and sufficient condition for raining is that the sky should be cloudy. (xii) I shall go to Agra, but I shall not see Taj Mahal. Solution. (i) Let p = I say, q = he refuses from that Statement is:p^q. (ii) p = I shall reach the station in time. q = I shall miss the train. Statement is:pvq. (iii) p = I shall go to Delhi, q = I shall see zoo. Statement is: p-q. (iv) p= he is ill, q = I shall wire. Statement is: pq. (v) p = I take leave, q = I fall ill. Statement is:-p^q. (vi) p= I shall be called, q = I shall remain here. Statement is: ~p q. (vii) p = Ram will go to work, q = Ram will remain in the house, r= Ram will white-wash the house. Statement is: pv (q^r). (viii) p = men were killed, q = women were killed, r= childern were killed. Statement is: p^(q^r). (ix) p= he will do labour, q = he will success, r= he will get pleasure. Statement is: (pq^qr)⇒ (pr). (x) p = teams do not arrive, q = whether is bad, r= there will be no match. Statement is: (pvq) →r. (xi) p = raining, q = cloudy sky. Statement is:pq. (xii) p = I shall go to Agra, q = I shall see Tajmahal. Statement is p⇒-q. Example 7. Whenever Ram and Shyam are present in the party then there is some trouble in the party. Today there is no trouble in the party. Hence Ram and Shyam are not present in the party. Write it in symbolic notations. Solution. p Ram and Shyam are present in the party, q = To be trouble in the party. Statement is:pq-q-p. Example 8. If p = It is 4 o'clock, q= the train is late, then state in words the following results: (i) pvq. (ii) p^q, (iii) p^(-q), (iv) qv-p. (v) (~p)^q, (vi) (p)^(-9), (vii) (-p) v (-9), (viii) ~ (p^q), (ix) - pp. Solution. (i) It is 4 o'clock or the train is late. (ii) It is 4 o'clock and the train is late. (iii) It is 4 o'clock and the train is not late. Or It is 4 o'clock but the train is not late. (iv) The train is late or it is not 4 o'clock. (v) It is not 4 o'clock and the train is late. (vi) It is not 4 o'clock and the train is not late. Or Neither it is 4 o'clock nor the train is late. (vii) It is not 4 o'clock or the train is not late. Or Either it is not 4 o'clock and the train is not late. (viii) It is not true that it is 4 o'clock and the train is late. (ix) If it is not 4 o'clock, then the train is late. Example 9. Let p = It is cold, q = It is raining. Give a simple verbal sentence which describes each of the following statements: (i) -p. (ii) p^q, (iii) pvq, (iv) qp, (v) p-q. (vi) qv-p, (vii) ~p^~q, (viii) p→~q, (ix) ~-q, (x) (p^~q) → p. (xi) ~~p, (xii) (p^~q) → q. Solution. (i) It is not cold. (ii) It is cold and raining. (iii) It is cold or it is raining. (iv) If it is cold, then it is raining. (v) If it is cold, then it is not raining. (vi) It is raining or it is not cold. (vii) It is not cold and it is not raining. (viii) It is cold if and only if it is not raining. (ix) It is not true that it is not raining. (x) If it is cold and not raining, then it is cold. (xi) It is not true that it is not cold. (xii) If it is raining and not cold, then it is raining. Example 10. If p = Ramesh is a player, q = Mohan is wise, then write the following symbols in sentences: (1) -pv-q. (ii) -pq. (iii) p^-q, (iv) p^q, (v) -(p^q). Solution. (i) Neither Ramesh is a player or nor Mohan is a wise boy. (ii) Ramesh is not a player iff Mohan is a wise boy. (iii) Ramesh is a player and Mohan is not a wise boy. (iv) Ramesh is a player and Mohan is a wise boy. (v) It is not true that Ramesh is a player and Mohan is a wise boy. ## 1.11. TRUTH TABLES The meaning and use of different logical connectives can be very well understood by the analysis of truth values. In symbols, true or false statements are expressed with the help of tables. These tables are called Truth tables. ## 1.12. BASIC LOGICAL OPERATIONS **A) Conjunction (A).** Suppose p and q are two statements. When two statements p, q are combined by using the word ‘and’, then we get a new statement represented by p^q. The resulting statement p^q is read as ‘p and q’ or ‘p meet q’. The statement p^q so obtained is called the conjunction of p and q. The conjunction of statements p and q i.e., p^q is true only in the condition when p and q both are true, i.e., (i) If p is true and q is true, then p^q is true. (ii) If p is true and q is false, then p^q is false. (iii) If p is false and q is true, then p^q is false. (iv) If p is false and q is false, then p^q is false. | P | q | p^q | |---|---|---| | T | T | T | | T | F | F | | F | T | F | | F | F | F | Note. The statements, used to form the resulting statement, are called components. For example, the components of resulting statement p^q are p and q. **(B) Disjunction (V).** Suppose p and q are two statements. When two statements p, q are joined by using the word ‘Or’, then we get a new statement represented by pvq. The resulting statement pvq is read as ‘p or q’ ‘p join q’. The statement pvq so obtained is called the disjunction of p and q. The disjunction of statements p and q i.e., pvq is false only in the condition when p and q both are false, i.e., (i) if p is true and q is true, then pv q is true, (ii) if p is true and q is false, then pvq is true, (iii) if p is false and q is true, then pvq is true, (iv) if p is false and q is false, then pvq is false. | P | q | pvq | |---|---|---| | T | T | T | | T | F | T | | F | T | T | | F | F | F | **(C) Negation (~).** Let p be a statement. The negation (or denial) of the statement p is obtained by putting the word ‘not’ and is represented by ‘p’ or by ‘~p’, -p is read as ‘not p’. Hence (i) if p is true, then p is false. (ii) if p is false, then p is true. Therefore the truth value of ~p is also true or false. | P | -P | |---|---| | T | F | | F | T | Thus the negation of a statement is a new statement which is the denial of the previous statement. But at the time of taking the negation (~) of a statement, one should be very cautious, which will be clear from the following example : Suppose p = Ramesh is a good player of Hockey, then -*p* never means ‘*Ramesh* is a bad player of Hockey’. -*p* has the following meaning: -*p* = *Ramesh* is not a good player of Hockey. **Remark.** The proposition p^q, pvq and~p are called atomic propositions. **(D) Conditional (⇒ or →).** Let p and q be any two statements. The statement of the type ‘if p then q’ is called conditional statement and is represented by ‘p⇒ q’. It is read as ‘p implies q’ ‘p’ is called antecedent and ‘q’ is called consequent. To be more clear about conditional statement, consider the following example : A father said to his son, "if you will stood first in the examination, then I shall purchase a scooter for you". If antecedent is denoted by p and consequent by q, then the given conditional statement is represented in symbols as ‘p⇒ q’ or as ‘p→ q’. Now there are following four possibilities: i.e.. (i) The son stands first in the examination and the father purchases a scooter for him, If p is true, q is true, then p⇒q is true. (ii) The son stands first in the examination and the father does not purchase a scooter for him, i.e., If p is true, q is false, then p⇒q is false. (iii) The son does not stand first in the examination and the father purchases a scooter for him, i.e., If p is false, q is true, then p⇒q is true. (iv) The son does not stand first in the examination and the father does not purchase a scooter for him, i.e., If p is false, q is false, then p⇒q is true. Hence we observe that the conditional statement is false only in the condition ‘when antecedent is true and consequent is false’. | P | q | p⇒q | |---|---|---| | T | T | T | | T | F | F | | F | T | T | | F | F | T | Note. Some authors use the symbol ‘→’ instead of the symbol ‘⇒’. **(E) Bi-Conditional (↔ or ≡).** Let pand q be two statements. The statement of the type ‘p⇒q and q⇒p’ i.e ‘p⇒q^q⇒p’ is called bi-conditional statement and it is represented by ‘p↔q’ or by ‘p≡q’. Bi-conditional statement p↔q is also read as follows : (a) ‘p if and only if q’, (b) ‘p then and only then when q’, (c) ‘p iff q’. To make more clear the bi-conditional statement, in example of (D) above, if the father had promised the son that, "I shall purchase a scooter for you if and only if you will stand first in the examination". It means that the father will not purchase a scooter for him if the son does not stand first in the examination. Thus case (ii) and (iii) both in above example are against his promise, so bi-conditional is a false statement. Hence we observe that a bi-conditional statement is true only in those cases when its both components have the same truth values. | P | q | p↔q | |---|---|---| | T | T | T | | T | F | F | | F | T | F | | F | F | T | Note. Let pand q be statement letters, then following are examples of statement patterns: (i) p^q, (ii) pvq, (iii) pvq- p, (iv) pq , (v) pq (vi) pvqqvp, **(F) NOR or Joint Denial (↓).** Let p and q be two statements. The statement of the type p↓ q, read as ‘Neither p nor q’ is called joint denial or NOR statement. In other words NOR (or joint denial) is the "negation of OR of two statements”. Thus p↓ q = ~(pvq). To make more clear the joint denial statement, in the example of (D) above, neither the father had promised to purchase a scooter nor the son has stood first in the examination. It means that p↓ q is true only when p and q both are false. | P | q | p↓q | |---|---|---| | T | T | F | | T | F | F | | F | T | F | | F | F | T | Example. Express the three connective v, and - in terms of the connective ↓. Solution. Truth tables for three connectives are as follows: (i) P| -P| P |---|---|---| | T | F | F | | F | T | T | -p = p↓p. (ii) P| q| p^q| P| P| q| q| (p↓p)↓ (q ↓ q) |---|---|---|---|---|---|---|---| | T | T | T| | T | F | F| | F | T | F| | F | F | F| p^q=(p↓p)^(q↓q). (iii) P | q | pvq| p | q | (p↓q)↓ (p ↓ q) |---|---|---|---|---|---|---| | T | T | T | | T | F | T | | F | T | T | | F | F | F | pvq = (p↓ q)↓ (p ↓ q). **(G) NAND (↑)** The connective NAND is denoted by the symbol ↑ and is defined as “negation of AND of two statements.” If p and q are two statements then their NAND is denoted by p ↑ q. Thus p↑q=~(p^q) Thus p↑ q is false only in the case when both p and q are true. | p | q | p ↑ q | |---|---|---| | T | T | F | | T | F | T | | F | T | T | | F | F | T | **(H) XOR (+)** The XOR of two statements is True if only if one of the two statements is true. In other words, if p and q are two statements then their XOR is denoted by p⊕q and is True if p is true or if q is true but not p and q both be true. | p | q | p+q | |---|---|---| | T | T | F | | T | F | T | | F | T| T | | F | F | F | Remark. The connection ↓ ↑ and ⊕ are called derived connections. Example. Construct the truth table for the propositions p ↑ q ↑r and p⊕q⊕r. Solution. | p | q | r | p ↑ q | p ↑ q ↑r| p⊕q | p ⊕ q⊕r| |---|---|---|---|---|---|---| | T | T | T | F | F | F | F | | T | T | F | F | F | F | T | | T| F | T | T | T | T | F | | T | F | F | T | T | T | T | | F| T | T | T | T | T | F | | F | T | F | T | T | T | T | | F | F | T | T | F | F | T | | F | F | F | T | T | F | F | ## 1.13. KINDS OF CONDITIONAL If p and q are any two statements, then some other conditional, related to conditional pq, are also used which are given in the following table. | Conditional | Name of kind | |---|---| | p⇒q | Direct implication | | q⇒p | Converse implication | | ~p⇒~q | Inverse or opposite implication | | ~q⇒~p | Contrapositive implication | Now we shall discuss contrapositive implication in detail. If p and q are two statements, then implications ~q⇒~p is called the contrapositive of implication p→ q, i.e., p→q and ~q→~p are contrapositive of each other. For example: if in any ∆ABC, ∠B = 90°, then BA² + BC² = AC². Its contrapositive will be as follows: In any ∆ABC, if BA² + BC² + AC2, then ∠B ≠ 90°. Again above both implications (i) and (ii) are contrapositive of each other. The following unified truth table, clearly explains the definitions of different kind of conditional: | P | q | -P | -q | p⇒q | q⇒p | ~p⇒~q | ~q⇒~p | |---|---|---|---|---|---|---|---| | T | T | F | F | T | T | T | T | | T | F | F | T | F | T | T | F | | F | T | T | F | T | F | F | T | | F | F | T | T | T | T | T | T | ## 1.14. TAUTOLOGY **Definition.** A tautology is a proposition which is true for all truth values of its sub propositions or components. A tautology is also called logically valid or logically true. Clearly in truth tables, all entries in the column of tautology are of ‘*T*’ only. ### ILLUSTRATIVE EXAMPLES Example 1. If p is a statement, prove that pp is a tautology. Solution. The truth table for p⇒p is as given here. Since all entries in the column of p⇒p are of ‘*T*’ only. Hence p⇒p is a tautology. | P | p⇒p | |---|---| | T | T | | F | T | Example. 2. (a) If p and q are two statements, prove that (p^q) →p is a tautology. Solution. Truth table for p^qp is as follows: | P | q | p^q | p^q⇒p | |---|---|---|---| | T | T | T | T | | T | F | F | T | | F | T | F | T | | F | F | F | T | Since all entries in the columns of p^q⇒p are T, it is a tautology. Example 2. (b) Prove that p^q⇒ q is a tautology. Solution. Truth table for p^qq. | P | q | p^q | p^q⇒q | |---|---|---|---| | T | T | T | T | | T | F | F | T | | F | T | F | T | | F | F | F | T | Note. Above 2(a) and 2(b) are called rules for simplification. Example 3. Prove that pv (-p) is a tautology. Solution. Truth table for pv (-p). | P | -P | pv-p | |---|---|---| | T | F | T | | F | T | T | Note. pv-pis a tautology. It shows any statements p is either true or false. Example 4. Laws of addition. Prove that: (i) ppvq, (ii) q⇒(pvq), are tautologies, when p and q are any two statements. Solution. Truth tables for p⇒pvq and qpvq are shown by a single truth table as given below : | P | q | pvq | p⇒pvq | q⇒pvq | |---|---|---|---|---| | T | T | T | T | T | | T | F | T | T | T | | F | T | T | T | T | | F | F | T | T | T | Since the columns under ppvq and qpvq contain only T’s. Hence given propositions are tautologies. Example 5. Prove that p^q⇒pvq is a tautology. Solution. Truth table for p^q⇒pvq: | P | q | p^q | pvq | p^q⇒pvq | |---|---|---|---|---| | T | T | T | T | T | | T | F | F | T | T | | F | T | F | T | T | | F | F | F | F | T | .. p^qpvqis tautology. Example 6. Is the statement (~p) v q is a tautology? Solution. Clearly (-p) v q is not a tautology from the following truth table : | P | -P | q | -Pvq | |---|---|---|---| | T | F | T | T | | T | F | F | F | | F | T | T | T | | F | T | F | T | Example 7. Prove that ~ {p^(-p)} is a tautology. Solution. Clearly the statement ~ {p^(~p)} is a tautology from the following truth table: | P | -P | p^(-p) | -{p^(-p)} | |---|---|---|---| | T | F | F | T | | F | T | F | T | Example 8. Prove that p⇒(~p)-p is a tautology. Solution. Truth table for p⇒ (-p)-p. | P | -P | p⇒~p | p⇒(~p)(~p) | |---|---|---|---| | T | F | F | T | | F | T | T | T | Hence the proposition p⇒ (~p)~p is a tautology. Example 9. Prove that (pq) →(~pvq) is a tautology. Solution. | P | q | -P | p⇒q | -pvq | p⇒q → (~pvq) | |---|---|---|---|---|---| | T | T | F | T | T | T | | T | F | F | F | F | T | | F | T | T | T | T | T | | F | F | T | T | T | T | Example 10. Prove that (pq) → (pq)^(qp) is a tautology. Solution. | P | q | p⇒q | q⇒p | (pq)^(qp) | (pq) ↔ (pq)^(qp) | |---|---|---|---|---|---| | T | T | T | T | T | T | | T | F | F | T | F | T | | F | T | T| F | F | T | | F | F | T | T | T | T| Example 11. Show that (pq)^(qp) → (pq) is a tautology. Solution. | P | q | p⇒q | q⇒p | (pq)^(qp) ⇒ p⇒q | |---|---|---|---|---| | T | T | T | T | T | | T | F | F | T | T | | F | T | T | F | T | | F | F | T | T | T | Example 12. Prove that (pq)^(qr) → (pr) is a tautology. Solution. For convenience, let pr=A and (pq)^(qr) = B | p | q | r | p⇒q | q⇒r | (suppose)| (suppose)| B= (pq)^(qr)| B→A= (pq)^(qr) → (pr) | |---|---|---|---|---|---|---|---|---| | T | T | T| T | T | T | T | T | T | | T | T | F | T | F | F | F | F | T | | T | F | T | F | T | F | F | F | T | | T | F | F | F | T | F | F | F | T | | F | T | T | T | T | T | T | T | T | | F | T | F | T | F | F | F | F | T | | F | F | T | T | T | T