Find a counterexample, if possible, to these universally quantified statements, where the domain
for all variables consists of all integers.
a) ∀x (x2 ≥x)
b)∀x(x>0∨x<0) c) ∀x (x = 1)
a:True,since x2−x⩾0for x∈(−∞,0][1,+∞)which includes all integersb:x=0−counterexamplec:x=0−counterexamplea:\\True, \sin ce\,\,x^2-x\geqslant 0 for\,\,x\in \left( -\infty ,0 \right] \left[ 1,+\infty \right) \\which\,\,includes\,\,all\,\,integers\\b:\\x=0-counterexample\\c:\\x=0-counterexamplea:True,sincex2−x⩾0forx∈(−∞,0][1,+∞)whichincludesallintegersb:x=0−counterexamplec:x=0−counterexample
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