the exercise asks: what is the value ( T or F ) of this proposition:
- $∃x, (2x+4=0\:\text{ is equivalent to }\:x=-2)$
Where x belongs to the natural numbers.
My train of thought:
- regardless of the values of x the equivalency is true
- however, I cannot determine a natural number that verifies this equivalence therefore I cannot state that the proposition is true.
- If I say this proposition is true then x belongs to Ø
- Can I do this?: $(∃x, 2x+4=0)$ is equivalent to $(∃x, x =-2) $ and therefore $F$ is equivalent to $F$ which is $T$.
$\exists x\in \Bbb N\, (2x+4=0\iff x=-2),$ partly unabbreviated, is $\exists x\,(x\in \Bbb N\land [2x+4=0 \iff x=-2]).$ Now $A\iff B$ is an abbreviation for $(A\land B)\lor (\,[\neg A]\land [\neg B]\,).$ So we have $$\exists x\,(x\in \Bbb N \land [(2x+4=0\land x=-2)\lor (2x+4\ne 0\land x\ne -2)]\,).$$ This is true if $\exists x\in \Bbb N $ is true because if $x\in \Bbb N$ then $(2x+4\ne 0\land x\ne -2).$