Images of intersections and unions

Task number: 2811

For the mapping \(f: A\to B\) and \(M,M'\subseteq A;\ N,N'\subseteq B\) replace \(\Box\) with the appropriate relation \(\subseteq, =\) or \(\supseteq\). If equality does not apply, characterize the mappings for which equality holds.

  • Variant 1

    \(f(M\cup M')\ \Box\ f(M) \cup f(M')\).

  • Variant 2

    \(f(M\cap M')\ \Box\ f(M) \cap f(M')\).

  • Variant 3

    \(f(M\setminus M')\ \Box\ f(M) \setminus f(M')\).

  • Variant 4

    \(f^{-1}(N\cup N')\ \Box\ f^{-1}(N) \cup f^{-1}(N')\).

  • Variant 5

    \(f^{-1}(N\cap N')\ \Box\ f^{-1}(N) \cap f^{-1}(N')\).

  • Variant 6

    \(f^{-1}(N\setminus N')\ \Box\ f^{-1}(N) \setminus f^{-1}(N')\).

Difficulty level: Moderate task
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