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Math Help - What is called this logical formula?

  1. #1
    Member
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    Aug 2008
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    What is called this logical formula?

    Hello,

    I want to know how the following formula in logic is called:
    (A\Rightarrow B)\Rightarrow ((A\wedge B)\Leftrightarrow A)

    Remarks:
    (1) I once read in a book ("Theory and Problems of Logic", McGraw-Hill) calling it "the absorption law".

    (2) In many textbooks, I see:
    "the absorption law"---- (A\wedge (A\vee B))\Leftrightarrow A
    "the idempotent law"---- (A\wedge A)\Leftrightarrow A

    (3) The formula in question is often found in set theory in the form:
    A\subset B\Rightarrow A\cap B = A

    Thank you for your time...
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  2. #2
    Junior Member
    Joined
    Oct 2006
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    Quote Originally Posted by wisterville View Post
    Hello,

    I want to know how the following formula in logic is called:
    (A\Rightarrow B)\Rightarrow ((A\wedge B)\Leftrightarrow A)

    Remarks:
    (1) I once read in a book ("Theory and Problems of Logic", McGraw-Hill) calling it "the absorption law".

    (2) In many textbooks, I see:
    "the absorption law"---- (A\wedge (A\vee B))\Leftrightarrow A
    "the idempotent law"---- (A\wedge A)\Leftrightarrow A

    (3) The formula in question is often found in set theory in the form:
    A\subset B\Rightarrow A\cap B = A

    Thank you for your time...
    It's a strengthened version of the following (usually derived) rule of inference, known as "absorption" in some texts.

    A -> B |- A -> (A ^ B)

    Clearly, strengthening the conclusion provides nothing new, since A ^ B |- A.
    (E.g., in an NDS this is just the primitive rule, caret elimination.)
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  3. #3
    Member
    Joined
    Aug 2008
    Posts
    80
    Thank you.

    I was confused because I saw the term "absorption " used in another context.

    Maybe it is viewed as a special version of
    "the absorption law"---- (A\wedge(A\vee B))\Leftrightarrow A
    if we take into account that
    A\Rightarrow B\vdash (A\vee B)\Leftrightarrow B.

    Any further comments are welcome.

    Bye.
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