{"id":2954,"date":"2025-06-06T07:21:08","date_gmt":"2025-06-06T07:21:08","guid":{"rendered":"https:\/\/diznr.com\/?p=2954"},"modified":"2025-06-06T07:21:08","modified_gmt":"2025-06-06T07:21:08","slug":"unless-plus-concept-previous-year-paper-discrete-mathematics-gate-2020-if-x-then-y-z-unless","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/unless-plus-concept-previous-year-paper-discrete-mathematics-gate-2020-if-x-then-y-z-unless\/","title":{"rendered":"Unless plus concept – previous year paper discrete mathematics- GATE 2025- If X then Y Unless Z"},"content":{"rendered":"

Unless plus concept – previous year paper discrete mathematics- GATE 2025- If X then Y Unless Z<\/p>\n

[fvplayer id=”187″]<\/p>\n

\u00a0“Unless” Concept in Discrete Mathematics – GATE 2025<\/strong><\/h3>\n

\u00a0Understanding “Unless” in Logic<\/strong><\/h4>\n

In propositional logic<\/strong>, “unless” can be rewritten using logical operators.<\/p>\n

The statement:<\/p>\n

“X\u00a0unless\u00a0Z”\\text{“X unless Z”}<\/span>“X\u00a0unless\u00a0Z”<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

means:<\/p>\n

\u00acZ\u2192X\\neg Z \\rightarrow X<\/span>\u00ac<\/span>Z<\/span>\u2192<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span><\/p>\n

(If Z is false, then X must be true.)<\/p>\n

Similarly, the statement:<\/p>\n

“If\u00a0X\u00a0then\u00a0Y\u00a0unless\u00a0Z”\\text{“If X then Y unless Z”}<\/span>“If\u00a0X\u00a0then\u00a0Y\u00a0unless\u00a0Z”<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

can be rewritten as:<\/p>\n

(Z\u2228X)\u2192Y(Z \\lor X) \\rightarrow Y<\/span>(<\/span>Z<\/span>\u2228<\/span><\/span>X<\/span>)<\/span>\u2192<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span><\/p>\n

This means “If Z is true OR X is true, then Y must be true.”<\/strong><\/p>\n

\u00a0Equivalent Logical Forms<\/strong><\/h3>\n
    \n
  1. \n

    “X unless Z”<\/strong><\/p>\n

    X\u2228ZX \\lor Z<\/span>X<\/span>\u2228<\/span><\/span>Z<\/span><\/span><\/span><\/span><\/span><\/p>\n

    Equivalent to:<\/p>\n

    \u00acZ\u2192X\\neg Z \\rightarrow X<\/span>\u00ac<\/span>Z<\/span>\u2192<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span><\/p>\n

    (If Z is false, then X must be true.)<\/p>\n<\/li>\n

  2. \n

    “If X then Y unless Z”<\/strong><\/p>\n

    (Z\u2228X)\u2192Y(Z \\lor X) \\rightarrow Y<\/span>(<\/span>Z<\/span>\u2228<\/span><\/span>X<\/span>)<\/span>\u2192<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span><\/p>\n

    Equivalent to:<\/p>\n

    \u00ac(Z\u2228X)\u2228Y\\neg (Z \\lor X) \\lor Y<\/span>\u00ac<\/span>(<\/span>Z<\/span>\u2228<\/span><\/span>X<\/span>)<\/span>\u2228<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span><\/p>\n

    Which simplifies to:<\/p>\n

    (\u00acZ\u2227\u00acX)\u2228Y(\\neg Z \\land \\neg X) \\lor Y<\/span>(<\/span>\u00ac<\/span>Z<\/span>\u2227<\/span><\/span>\u00ac<\/span>X<\/span>)<\/span>\u2228<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n

    \u00a0GATE 2025 Previous Year Question on “Unless”<\/strong><\/h3>\n

    Question:<\/strong><\/h4>\n

    Which of the following is logically equivalent to:<\/p>\n

    “If\u00a0X\u00a0then\u00a0Y\u00a0unless\u00a0Z”\\text{“If X then Y unless Z”}<\/span>“If\u00a0X\u00a0then\u00a0Y\u00a0unless\u00a0Z”<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    (A) (X\u2192Y)\u2228Z(X \\rightarrow Y) \\lor Z<\/span>(<\/span>X<\/span>\u2192<\/span><\/span>Y<\/span>)<\/span>\u2228<\/span><\/span>Z<\/span><\/span><\/span><\/span>
    (B) X\u2228(Y\u2228Z)X \\lor (Y \\lor Z)<\/span>X<\/span>\u2228<\/span><\/span>(<\/span>Y<\/span>\u2228<\/span><\/span>Z<\/span>)<\/span><\/span><\/span><\/span>
    (C) (Z\u2228X)\u2192Y(Z \\lor X) \\rightarrow Y<\/span>(<\/span>Z<\/span>\u2228<\/span><\/span>X<\/span>)<\/span>\u2192<\/span><\/span>Y<\/span><\/span><\/span><\/span>
    (D) \u00acZ\u2228(X\u2192Y)\\neg Z \\lor (X \\rightarrow Y)<\/span>\u00ac<\/span>Z<\/span>\u2228<\/span><\/span>(<\/span>X<\/span>\u2192<\/span><\/span>Y<\/span>)<\/span><\/span><\/span><\/span><\/p>\n

    Solution Approach:<\/strong><\/h4>\n

    We break down:<\/p>\n

    “If\u00a0X\u00a0then\u00a0Y\u00a0unless\u00a0Z”\\text{“If X then Y unless Z”}<\/span>“If\u00a0X\u00a0then\u00a0Y\u00a0unless\u00a0Z”<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n