GATE | PYQ | Discrete Mathematics |2026

Let 𝑅 be a binary relation on the set {1, 2, … , 10}, where (𝑥, 𝑦) ∈ 𝑅 if the product of 𝑥 and 𝑦 is square of an integer. Which of the following properties is/are satisfied by 𝑅? [MSQ || GATE 2026 Set-2 || 1 Marks ]

For two different persons 𝑥 and 𝑦, the predicate 𝑀(𝑥, 𝑦) denotes that x knows y. Consider the following statement.

There is a person who does not know anyone else, but that person is known by everyone else.

Which one of the following expressions represents the above statement? [MCQ || GATE 2026 Set-2 || 1 Marks ]

Let 𝐺(𝑉, 𝐸) be a simple, undirected graph. A vertex cover of 𝐺 is a subset 𝑉 ′ ⊆ 𝑉 such that for every (𝑢, 𝑣) ∈ 𝐸, 𝑢 ∈ 𝑉 ′or 𝑣 ∈ 𝑉 ′ . Let the size of the smallest vertex cover in 𝐺 be 𝑘. Let 𝑆 be any vertex cover of size 𝑘.
For a vertex 𝑣 ∈ 𝑉, which of the following constraints will always ensure that 𝑣 ∈ 𝑆 ? [MCQ || GATE 2026 Set-2 || 2 Marks ]

Let 𝐺 be an undirected graph, which is a path on 8 vertices. The number of matchings in 𝐺 is ______. (answer in integer) [NAT || GATE 2026 Set-1 || 2 Marks ]

Consider a complete graph 𝐾𝑛 with 𝑛 vertices (𝑛 > 4). Note that multiple spanning trees can be constructed over 𝐾𝑛. Each of these spanning trees is represented as a set of edges. The Jaccard coefficient between any two sets is defined as the ratio of the size of the intersection of the two sets to the size of the union of the two sets. Which one of the following options gives the lowest possible value for the Jaccard coefficient between any two spanning trees of 𝐾𝑛? [MCQ || GATE 2025 Set-2 || 2-marks ]