mcs 013 solved assignment 2017-18

MCS - 013 

Discrete Mathematics

 

(a) Explain different logical connectives with the help of examples.

 

In logic, a logical connective (also called a logical operator) is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective.
The most common logical connectives are binary connectives (also called dyadic connectives) which join two sentences which can be thought of as the function's operands. Also commonly, negation is considered to be a unary connective.
Logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic. Semantics of a logical connective is often, but not always, presented as a truth function.
A logical connective is similar to but not equivalent to a conditional operator.^[1]


In the grammar of natural languages two sentences may be joined by a grammatical conjunction to form a grammatically compound sentence. Some but not all such grammatical conjunctions are truth functions. For example, consider the following sentences:

A: Jack went up the hill.

B: Jill went up the hill.

C: Jack went up the hill and Jill went up the hill.

D: Jack went up the hill so Jill went up the hill.

The words and and so are grammatical conjunctions joining the sentences (A) and (B) to form the compound sentences (C) and (D). The and in (C) is a logical connective, since the truth of (C) is completely determined by (A) and (B): it would make no sense to affirm (A) and (B) but deny (C). However, so in (D) is not a logical connective, since it would be quite reasonable to affirm (A) and (B) but deny (D): perhaps, after all, Jill went up the hill to fetch a pail of water, not because Jack had gone up the hill at all.
Various English words and word pairs express logical connectives, and some of them are synonymous. Examples are:

Word	Connective	Symbol	Logical Gate
and	conjunction	"∧"	AND
and then	conjunction	"∧"	AND
and then within	conjunction	"∧"	AND
or	disjunction	"∨"	OR
either, but not both	exclusive disjunction	"⊻"	XOR
implies	implication	"→" "←"
if...then	implication	"→" "←"
if and only if	biconditional	"↔"	XNOR
only if	implication	"→" "←"
just in case	biconditional	"↔"	XNOR
but	conjunction	"∧"	AND
however	conjunction	"∧"	AND
not both	alternative denial	"|"	NAND
neither...nor	joint denial	"↓"	NOR
not	negation	"¬"	NOT
it is false that	negation	"¬"	NOT
it is not the case that	negation	"¬"	NOT
although	conjunction	"∧"	AND
even though	conjunction	"∧"	AND
therefore	implication	"→" "←"
so	implication	"→" "←"
that is to say	biconditional	"↔"	XNOR
furthermore	conjunction	"∧"	AND