Assignment #2, Question 7

In this question we are required to use symbols to represent the truth values of phrases, and to create new (connective) symbols in cases where using {and, or, not} might be too cumbersome.

New Symbols:

Let "<=>" = x IFF y = NOT[(x or y) and NOT(x and y)]

Let "<=" = x ONLY IF y = NOT(y) or x

Let "XOR" = x OR y AND NOT(x AND y)

Statements:

Poison cased the victim's death if and only if there was a change in his blood chemistry or a residue of poison in his stomach.

Let
A=Poison cased the victim's death
B=there was a change in his blood chemistry
C=residue of poison in his stomach

A <=> (B or C) {I believe we don't use XOR here because if both a change in blood chemistry and residue of poison is observed, then Poison still caused the victim's death. So no need for XOR here, just OR.}

There was neither a change in blood chemistry nor a residue of poison in his stomach, but there were puncture marks on the body.

Let D=there were puncture marks on the body.

NOT (B OR C) AND D

Poison was injected by a needle only if there were puncture marks on the body.

Let E=poison was injected by a needle

E <= D

Let S=Either poison was the cause of the victim's death, or there are no puncture marks on the body.

I initially thought the truth equivalent should be A OR NOT(D) but this was merely a 'straight' translation of the sentence instead of a truth equivalent statement. The 'Either' at the start of the sentence throws everything off and is a hint that S is an 'exclusive or' (XOR) case, where "you can have one or the other but not both events happening".

So S can be written as A XOR NOT(D).