# Current Affairs JEE Main & Advanced

#### Boolean Functions

Definition : Any expression like $x\wedge x'$, $a\wedge b'$, $[a\wedge (b\vee c')]\vee (a'\wedge b'\,\wedge c)$ consisting of combinations by $\vee$ and $\wedge$ of finite number of elements of a Boolean Algebra B is called a boolean function.   Let $B=\{a,b,c,....\}$ be a boolean algebra by a constant we mean any symbol as 0 and 1, which represents a specified element of B.   By a variable we mean a symbol which represents a  arbitrary element of B   If in the expression ${x}'\vee (y\wedge z)$ we replace $\vee$ by  + and $\wedge$ by ., we get ${x}'+y.z$. Here ${x}'$ and $y\wedge z$ are called monomials and the whole expression ${x}'\vee (y\wedge z)$ is called a polynomial.

#### Introduction

Boolean algebra is a tool for studying and applying mathematical logic which was originated by the English mathematician George Boolean. In 1854 he wrote a book “An investigation of the law of thought”, be developed a theory of logic using symbols instead of words. This more algebraic treatment of subject is now called boolean algebra     Definition : A non empty set B together with two operations denoted by $'\vee '$ and $\,'\wedge '$  is said to be a boolean algebra if the following axioms hold :   (i) For all $x,y\in B$   (a) $x\vee y\in B$                          (Closure property for $\vee$)   (b) $x\wedge y\in B$                   (Closure property for $\wedge$)   (ii) For all $x,y\in B$   (a) $x\vee y=y\vee x$                 (Commutative law for $\vee$)   (b) $x\wedge y=y\wedge x$    (Commutative law for $\wedge$)   (iii) For all x, y and z in B,   (a) $(x\vee y)\vee z=x\vee (y\vee z)$ (Associative law of$\vee$)   (b) $(x\wedge y)\wedge z=x\wedge (y\wedge z)$       (Associative law of $\wedge$)   (iv) For all x, y and z in B,   (a) $x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)$           (Distributive law of $\vee$ over$\wedge$)   (b) $x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z)$     (Distributive law of $\wedge$ over $\vee$)   (v) There exist elements denoted by 0 and 1 in B such that for all $x\in B$,   (a) $x\vee 0=x$                                                                                           (0 is identity for $\vee$)   (b) $x\wedge 1=x$                                                                                  (1 is identity for $\wedge$)   (vi) For each $x\in B$, there exists an element denoted by x¢, called the complement or negation of x in B such that   (a) $x\vee x'=1$   (b) $x\wedge x'=0$                                                                                    (Complement laws)   Principle of duality   The dual of any statement in a boolean aglebra B is the statement obtained by interchanging the operation Ú and Ù, and simultaneously interchanging the elements 0 and 1 in the original statement.   In a boolean algebra, the zero element 0 and the unit element 1 are unique.   Let B be a boolean algebra.  Then, for any x and y in B, we have   (a) $x\vee x=x$                   more...

#### Duality

Definition : Two compound statements ${{S}_{1}}$ and ${{S}_{2}}$ are said to be duals of each other if one can be obtained from the other by replacing $\wedge$by $\vee$ and $\vee$by $\wedge$.
• The connective $\wedge$and $\vee$are also called duals of each other

• If a compound statements contains the special variable
t (tautology) or c (contradiction),  then to obtain its dual we replace t by c and c by t in addition to replacing $\wedge$by $\vee$and $\vee$by $\wedge$.
• Let $S(p,q)$be a compound statement containing two sub- statements and ${{S}^{*}}$(p, q) be its dual. Then,
(i) $\tilde{\ }S(p,q)\equiv {{S}^{*}}(\tilde{\ }p,\tilde{\ }q)$   (ii) $\tilde{\ }{{S}^{*}}(p,q)\equiv S(\tilde{\ }p,\tilde{\ }q)$
• The above result can be extended to the compound statements having finite number of sub- statements. Thus, if $S({{p}_{1}},{{p}_{2}},....{{p}_{n}})$ is a compound statement containing n sub-statement ${{p}_{1}},{{p}_{2}},....,{{p}_{n}}$ and ${{S}^{*}}({{p}_{1}}{{p}_{2}},....,{{p}_{n}})$ is its dual. Then,
(i) $\tilde{\ }S({{p}_{1}},{{p}_{2}},....,{{p}_{n}})\equiv {{S}^{*}}(\tilde{\ }{{p}_{1}},\tilde{\ }{{p}_{2}},....,\tilde{\ }{{p}_{n}})$   (ii) $\tilde{\ }{{S}^{*}}({{p}_{1}},{{p}_{2}},....,{{p}_{n}})\equiv S(\tilde{\ }{{p}_{1}},\tilde{\ }{{p}_{2}},....,\tilde{\ }{{p}_{n}})$

#### Algebra of Statements

In the previous section, we have seen that statements satisfy many standard results. In this section, we shall state those results as laws of algebra of statements.   The following are some laws of algebra of statements.   (i) Idempotent laws : For any statement p, we have   (a) $p\vee p\equiv p$                                   (b) $p\wedge p\equiv p$   (ii) Commutative laws : For any two statements p and q, we have   (a) $p\vee q\equiv q\vee p$                      (b) $p\wedge p\equiv q\wedge p$   (iii) Association laws : For any three statements $p,q,r$  we have   (a) $(p\vee q)\vee r\equiv p\vee (q\vee r)$      (b) $(p\wedge q)\wedge r\equiv p\wedge (q\wedge r)$   (iv) Distributive laws : For any three statements $p,q,r$ we have   (a) $p\wedge (p\vee q)\equiv (p\wedge q)\vee (q\wedge r)$   (b) $p\vee (p\wedge q)\equiv (p\vee q)\wedge (q\vee r)$   (v) Demorgan’s laws : If p and q are two statements, then   (a) $\tilde{\ }(p\wedge q)\equiv \tilde{\ }p\vee \tilde{\ }q$        (b) $\tilde{\ }(p\vee q)\equiv \tilde{\ }p\wedge \tilde{\ }q$   (vi) Identity laws : If t and c denote a tautology and a contradiction respectively, then for any statement p, we have   (a) $p\wedge t\equiv p$              (b) $p\vee c\equiv p$   (c) $p\vee t\equiv t$    (d) $p\wedge c\equiv c$   (vii) Complement laws : For any statements p, we have   (a) $p\vee \tilde{\ }p=t$   (b) $p\wedge \tilde{\ }p=c$   (c) $\tilde{\ }t=c$     (d) $\tilde{\ }c=t$   where t and c denote a tautology and a contradiction respectively.   (viii) Law of contrapositive : For any two statements p and q, we have   $p\Rightarrow q\equiv \,\tilde{\ }q\Rightarrow \,\tilde{\ }p$   (ix) Involution laws : For any statement p, we have $\tilde{\ }(\tilde{\ }p)\equiv p$

Let $p,q,r,....$ be statements, then any statement involving $p,q,r$,....and the logical connectives $\wedge ,\vee ,\tilde{\ },\Rightarrow ,\Leftrightarrow$ is called a statement pattern or a Well Formed Formula (WFF).   For example   (i) $p\,\vee \,q$   (ii)  $p\Rightarrow q$   (iii) $((p\wedge q)\vee r)\Rightarrow (s\wedge \tilde{\ }s)$   (iv) $(p\Rightarrow q)\Leftrightarrow (\tilde{\ }q\Rightarrow \tilde{\ }p)$etc.   are statement patterns.   A statement is also a statement pattern.   Thus, we can define statement pattern as follows.   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.
• The negation of a tautology is a contradiction and vice versa.

#### Negation of Compound Statements

We have learnt about negation of a simple statement. Writing the negation of compound statements having conjunction, disjunctions, implication, equivalence, etc, is not very simple. So, let us discuss the negation of compound statement.   (i) Negation of conjuntion :   If p and q are two statements, then $\tilde{\ }(p\wedge q)\equiv (\tilde{\ }p\,\vee \tilde{\ }q)$   (ii) Negation of disjuntion :   If p and q are two statements, then $\tilde{\ }(p\vee q)\equiv (\tilde{\ }p\,\wedge \tilde{\ }q)$   (iii) Negation of implication :   If p and q are two statements, then $\tilde{\ }(p\Rightarrow q)=(p\,\wedge \tilde{\ }q)$             (iv) Negation of biconditional statement or equivalence :   If p and q are two statements, then   $\tilde{\ }(p\Leftrightarrow q)=(p\wedge \tilde{\ }q)\vee (q\wedge \tilde{\ }p)$

#### Logical Equivalence

Logically equivalent statement : Two compound ${{S}_{1}}(p,q,r,...)$ and ${{S}_{2}}(p,q,r...)$ are said to be logically equivalent, or simply equivalent if they have the same truth values for all logically possibilities.   If statements ${{S}_{1}}(p,q,r,...)$ and ${{S}_{2}}(p,q,r...)$ are  logically equivalent, then we write  ${{S}_{1}}(p,q,r,...)\equiv {{S}_{2}}(p,q,r...)$   It follows from the above definition that two statements ${{S}_{1}}$ and ${{S}_{2}}$ are logically equivalent if they have identical truth tables i.e., the entries in the last column of the truth tables are same.

#### Basic Logical Connectives or Logical Operators

Definition : The phrases or words which connect simple statements are called logical connectives or sentential connectives or simply connectives or logical operators.   In the following table, we list some possible connectives, their symbols and the nature of the compound statement formed by them.
 Connective Symbol Nature of the compound statement formed by using the connective and $\wedge$ Conjunction or $\vee$ disjunction If....then $\Rightarrow$or $\to$ Implication or conditional If and only if (iff) $\Leftrightarrow$ or $\leftrightarrow$ Equivalence or bi-conditional not $\tilde{\ }$ or $\neg$ Negation
(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 statements, then $p\wedge q$ denotes the conjunction of p and q and is read as “p and q”. more...

#### Truth Tables

Definition : 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-statement  $p,q,r,\text{ }.....$ etc, is called the truth table of statement S.   Construction of truth table : In order to construct the truth table for a compound statement, we first prepare a table consisting of rows and columns. At the top of the initial columns, we write the variables denoting the sub-statements or constituent statements and then we write their truth values, in the last column. We write the truth value of the compound statement on the basis of the truth values of the constituent statements written in the initial columns. If a compound statement is made up of two simple statement, then the number of rows in the truth table will be ${{2}^{2}}$ and if it is made up of three simple statements, then the number of rows will be ${{2}^{3}}$. In general, if the compound statement is made up of n sub-statements, then its truth table will contain ${{2}^{n}}$ rows.

#### Types of Statements

In Mathematical logic, we generally come across two types of statements or proposition, namely, simple statements and compound statements as defined below.   (i) Simple statements : Any statement or proposition whose truth value does not explicity depend on another statement is said to be a simple statement.   In other words, a statement is said to be simple if it cannot be broken down into simpler statements, that is, if it is not composed of simpler statements.   (ii) Compound statements : If a statement is combination of two or more simple statements, then it is said to be a compound statement or a compound proposition.

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