Find the truth

Algebra Level 5

This is a part of set Find the Truth

\(x\) and \(y\) are non-zero reals with \(x > y\).

\( (I)\quad \sin { x } \ge \sin ^{ 2 }{ x } \\ (II)\quad \sin { |x| } >\sin ^{ 2 }{ |x| } \\ (III)\quad { x }^{ x }\quad is\quad real\\ (IV)\quad { |x| }^{ x }\quad is\quad real\\ (V)\quad { |x| }^{ |x| }\quad is\quad real\\ (VI)\quad { x }^{ |x| }\quad is\quad real\\ (VII)\quad logx\quad is\quad real\\ (VIII)\quad log|x|\quad is\quad real\\ (IX)\quad \frac { x }{ y } \quad is\quad rational\\ (X)\quad \sqrt { { x }^{ 2 }-{ y }^{ 2 } } \quad is\quad real\\ \\ \\ Find\quad the\quad sum\quad of\quad the\quad serial\quad numbers\quad of\quad the\quad statements\\ that\quad are\quad definitely\quad true.\quad Consider\quad infinities\quad to\quad be\quad real.\\ \\ E.g.\quad If\quad statements\quad II\quad and\quad III\quad are\quad true,\quad then\quad answer\quad is\quad 2+3=5\\ If\quad none\quad of\quad the\quad statements\quad are\quad true,\quad answer\quad is\quad 0.\)

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