Question

Show that these statements are inconsistent: “If Sergei takes the job offer then he will get a signing bonus.” “If Sergei takes the job offer, then he will receive a higher salary.” “If Sergei gets a signing bonus, then he will not receive a higher salary.” “Sergei takes the job offer.”

Short answer

Interpretation of symbolsInterpretation of symbols

Step-by-step solution

Interpretation of symbols

Negation\(\neg p\): “not p.”

Disjunction\(p \vee q\): p or q

Conjunction\(p \wedge q\): p and q

Exclusive or\(p \oplus q\): p or q, but not both

Conditional statement\(p \to q\): if p, then q

Bi-conditional statement\(p \leftrightarrow q\): p, if and only if q

Existential quantification\(\exists xP\left( x \right)\): There exist an element x in the domain such that P(x).

Universal quantification\(\forall xP\left( x \right)\): P(x) for all values of x in the domain.

Proof for the given statements to be inconsistent

Let us assume:

Translate the given statements using the interpretations. The statements are inconsistent, if one variable is both true and false is obtained (which of course cannot be true).

	Step	Reason
1.	\(p \to q\)	Premise
2.	\(p \to r\)	Premise
3.	\(q \to \neg r\)	Premise
4.	\(p\)	Premise
5.	\(q\)	Modus ponens from (1) and (4)
6.	\(r\)	Modus ponens from (2) and (4)
7.	\(\neg r\)	Modus ponens from (3) and (5)

It is obtained that \(r\)has to be true and \(\neg r\) has to be true. However, \(\neg r\) is true when \(r\)is false. Thus,it is obtained that \(r\)is both true and false, which is impossible and thus the statements are inconsistent.

Therefore, the statements are inconsistent.