Question

1. Is the following set of connectives functionally complete? Justify your answer.{∧,∨,→,↔}

Answer 1

Ans: NO

Justification: Logical connectivities is a set of 5 basic connectivities are as follows :

NAME OF CONNECTIVE	CONNECTIVE WORD	SYMBOL
NEGATION	NOT	"⌉" or "∼" or ‘ or "–"
CONJUNCTION	AND	∧
DISJUNCTION	OR	∨
CONDITIONAL	if-then	→
BICONDITIONAL	if and only if	↔
BRACKETS		( )

(1): Negation: It changes the probability from "True to False" and from "False to True" use as with any proposition

Ex: If X is a proposition, then the negation of X is a proposition which is:

Truth Table

X	~X
TRUE	FALSE
FALSE	TRUE

Means: X= Raining Outside, ~X= Not Raining Outside.

(2): Conjunction: If X and Y are two propositions, then the conjunction of X and Y is a proposition which is :

Example:

• If both the propositions are True then True
• Both are False then false.

Truth Table

X	Y	X∧Y
TRUE	TRUE	TRUE
TRUE	FALSE	FALSE
FALSE	TRUE	FALSE
FALSE	FALSE	FALSE

X=1+2, Y= Raining Outside

The conjunction of X and Y are X∧Y= 3 and Raining Outside.

(3):Disjunction: If X and Y are two propositions, then the Disjunction of X and Y is a proposition which is :

Example:

• True when either one of X or Y or both are True.
• False when both X and Y are False

Truth Table

X	Y	X∨Y
TRUE	TRUE	TRUE
TRUE	FALSE	FALSE
FALSE	TRUE	FALSE
FALSE	FALSE	FALSE

X=1+2, Y= Raining Outside

The Disjunction of X and Y are X∧Y= 3 or Raining Outside.

(4): Conditional: If X and Y are two propositions, then:

• "If X then Y" is called a conditional or implication propositions.
• It is true when both X and Y are or when X is false.
• It is false when X is true and Y is false.

Truth Table

X	Y	X→Y
F	F	T
F	T	T
T	F	F
T	T	F

if a=b and b=c then a=c

(5): Biconditional: If X and Y are two propositions, then:

• X if and only if Y is called a biconditional or bi-implication proposition.
• it is true when either both X and Y are true or both X and Y are false.
• it is false in all other cases.

Truth Table

X	Y	X↔Y
F	F	T
F	T	F
T	F	F
T	T	T

Example:

• He goes to play a match if and only if it does not rain.
• Birds fly if and only if the sky is clear.