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GATE CSE 2025 Session 2 Q42: Pointwise order on functions from {1..n} to {0,1}

Question

Let F\mathcal{F} be the set of all functions from {1,…,n}\{1, \ldots, n\} to {0,1}\{0, 1\}. Define the binary relation ≼\preccurlyeq on F\mathcal{F} as follows:

∀f,g∈F,f≼g if and only if ∀x∈{1,…,n}, f(x)≤g(x), where 0≤1.\forall f, g \in \mathcal{F},\quad f \preccurlyeq g \text{ if and only if } \forall x \in \{1, \ldots, n\},\ f(x) \le g(x), \text{ where } 0 \le 1.

Which of the following statement(s) is/are TRUE?

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Answer: B, C (official GATE 2025 final answer key)

Concept used: Functions to {0,1}\{0,1\} are subsets of {1,…,n}\{1,\ldots,n\}, and the pointwise order is set inclusion.

First step: Identify ff with {x:f(x)=1}\{x : f(x) = 1\}; then ≼\preccurlyeq is ⊆\subseteq, with pointwise max and min as join and meet.

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Source: official GATE 2025 Session 2 CS question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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