what type of a relation is less than in the set of real numbers
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It’s not reflexive. In fact x<xx<x is false for all x∈Rx∈R, thus it is irreflexive.
It’s anti-symmetric but not in a usual way. The statement of anti-symmetry is R(a,b)∧R(b,a)⟹b=aR(a,b)∧R(b,a)⟹b=a, for a relation RR. But, for <<, the LHS is never true i.e. a<b∧b<aa<b∧b<a is always false. Thus, the statement of anti-symmetry is true, because if AAis false then A⟹BA⟹B is trivially true.
Irreflexive and anti-symmetric is then termed as asymmetric.
That it is transitive is clear.
It’s anti-symmetric but not in a usual way. The statement of anti-symmetry is R(a,b)∧R(b,a)⟹b=aR(a,b)∧R(b,a)⟹b=a, for a relation RR. But, for <<, the LHS is never true i.e. a<b∧b<aa<b∧b<a is always false. Thus, the statement of anti-symmetry is true, because if AAis false then A⟹BA⟹B is trivially true.
Irreflexive and anti-symmetric is then termed as asymmetric.
That it is transitive is clear.
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1
It’s not reflexive. In fact x<xx<x is false for all x∈Rx∈R, thus it is irreflexive.
It’s anti-symmetric but not in a usual way. The statement of anti-symmetry is R(a,b)∧R(b,a)⟹b=aR(a,b)∧R(b,a)⟹b=a, for a relation RR. But, for <<, the LHS is never true i.e. a<b∧b<aa<b∧b<a is always false. Thus, the statement of anti-symmetry is true, because if AA is false then A⟹BA⟹B is trivially true.
Irreflexive and anti-symmetric is then termed as asymmetric.
That it is transitive is clear
It’s anti-symmetric but not in a usual way. The statement of anti-symmetry is R(a,b)∧R(b,a)⟹b=aR(a,b)∧R(b,a)⟹b=a, for a relation RR. But, for <<, the LHS is never true i.e. a<b∧b<aa<b∧b<a is always false. Thus, the statement of anti-symmetry is true, because if AA is false then A⟹BA⟹B is trivially true.
Irreflexive and anti-symmetric is then termed as asymmetric.
That it is transitive is clear
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