Understanding of conditions

Rules for the compound statement with ‘Or’

  • A compound statement with an ‘Or’ is true when one component statement is true or both are true.
  • A compound statement with an ‘Or’ is false when both the component statements are false.

Quantifiers

Quantifiers are phrases like “There exists” and “For all”. A word closely connected with “there exists” is “for every” (or for all). Hence, the words “And” and “Or” are called connectives and “There exists” and “For all” are called quantifiers.

Axiom of disjunction:

If p, q are propositions, then p Ú q is true if p is true or q is true or both true, and it is false if both p & q are false.

Disjunction table

(iv) Conditional or implication (Þ):

If p and q be two statements, then p Þ q means ‘p implies q’. This type of statements are called conditional statements.

Example:

            p : x2 = 4

            q : x = ± 2

Axiom of conditional:

A conditional p Þ q is false if p is true and q is false, otherwise true.

Here p is called hypothesis and q is called conclusion.

(v) Biconditional (Û):

If p and q are two statements, then p Û q means ‘p if and only if q’. This type of statements are called Biconditional statements.

Example:

            p : Five is a prime number.

            q : It has no proper divisor.

            P Û q : Five is prime if and only if it has no proper divisor

Biconditional table

 

Truth Table:

The tabular classification of complex statements given above are taken as axioms upon which logical decissions are based and such tables are called truth tables.

Given a conditional p ® q, three other propositions related to itcanbe framed.

  1. Converse : q ® p
  2. Inverse : ~ p ® ~ q
  3. Contra positive : ~® ~ p.

 

Fallacy:           A statement which is always false is called fallacy.

Tautology:      A statement which is always true is called tautology.

Equivalent:    Two statements are equivalent if their truth values are identical i.e. entries in the two columns are exactly same.

Example:         p  ~ (~p) are equivalent.

Some important tautologies are given below.

(i) Law of addition:

If p & q are two statements, then

p Þ p Ú q,   q Þ p Ú q

(ii) Law of Implication: 

 p Ù q Þ p,   p Ù q Þ = q, where p & q are two statements.

(iii) Commutative laws:

If p & q are two statements, then p Ù q Û q Ù q, p Ú q Û q Ú p

(iv) Associative laws:

If p, q & r are two statements, then

p Ù  (q Ù r) Û (p Ù q) Ù r

p Ú (q Ú r) Û (p Ú q) Ú r

(v) Distributive laws:

If p, q and r are statements, then

    1. pÙ(qÚr)Û(pÙq) Ú (p Ù r)
    2. pÚ(q Ùr)Û(pÚq) Ù (p Ú r)
Truth table of ‘a’
Truth table of ‘b’

(vi) Idempotent Laws:

If p is a statement, then

p Ú p Û p, p Ù p Û p

(vii) Law of double negation:

If p is a statement, then p Û ~ (~p)

(viii) Law of contra positive:

If p, q are two statements, then (p Þ q) Û (~ q Þ ~ p)

(ix) Law of inclusive middle:

If p is a statement, then p Ú ~ p

(x) De-morgans laws:

If p & q are two statements, then

  1. ~  (p Ù q) Û (~ p Ú ~ q)
  2. ~ (p Ú q) Û ~ p Ù ~ q

These are familiar examples of tautologies.

Compound statements

Many mathematical statements are obtained by combining one or more statements using some connecting words like “and”, “or”, etc.

Now, suppose two statements are given as below:

p: 5 is an odd number.

q: 5 is a prime number.

We can combine these two statements with “and”.

r: 5 is both an odd and prime number.

This is a compound statement.

From this, we can write the following definition:

Examples

 

Question: Prove that ~ (pÙq) = ~ p Ú ~ q by constructing a truth table.

Solution:

This can be proved by constructing the following truth table.

 Since the truth values of column 6 & 7 are same it proves that

            ~ (pÙq) = ~ p Ú ~ q

Question: Prove that {(p®q) Ù (q ® r)} ® (p ® r) is a tautology by constructing the corresponding truth table.

Solution:

Question: If r is any false statement then show that p ®r  is logically equivalent to p Ù (~ q) ® r

Solution:

Here we observe that the truth values of column 4 and column 7 are equal in the above truth table, it proves that p ®q is logically equivalent to

p Ù (~ q) ® r.

Solution:

Construct the truth table of [p ® (pÚq)]  ® [q  ® (pÙq)]

Hence or otherwise decide, we either it is a tautology or not.

Question: Examine whether  ~ q Ú p and q ® p are equivalent.

Solution:

Let’s form a truth table of ~ q Ú p  and q ® p to decide weather they are equivalent or not.

As the truth values of column 4 & 5 are same, hence they are equivalent.

Question: Constructing the truth table, state whether the following statement is a tautology  not.

 (pÙq) Ú (~p Ú ~ q)

As all the truth values of column 7, are true. Hence the given statement is a tautology.

Question: Draw the truth table of the following statement.

~ (p® q) « [(~ p Ú q) « ~ q]  

Or

Contrapositive and converse

Contrapositive and converse are certain other statements which can be formed from a given statement with “if-then”.

  • The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒ ∼p .
  • The converse of a statement p ⇒ q is the statement q ⇒ p.
  • p ⇒ q together with its converse, gives p if and only if q

Solution: