Out of the integers 1 to 1000. How many of them are neither divisible by 3, nor by 5 nor by 7?
Define partition of a set. Let a relation R is defined by xRy iff (x-y) is divisible by 5. Then show that R is an equivalence relation. Also, find the all distinct partition for R.
Define the relation divisibility on the set D12. where D12 is the set of all positive divsors of 12.(i) show that it is an partial orderd relation
Show that the function from R to R defined by f(x)=4x+8 is one-one and onto. Also, find the inverse.
Define the isomorphic graphs. Show that following two graphs are isomorphic with defined one-one correspondence between vertices.
Find the minimum number of students needed to guarantee that five of them belong to the same class.
Define integral domain. show that the set Z6={[0], [1], [2], [3], [4], [5]} is not an integral domain.
Use warshall's algorithm to find the path between each pair of vertices for given directed graph
Show that the set Z5={[0],[1],[2],[3],[4]} is the field under the operation addition modulo 5 and multiplication modulo 5.
Find the inverse, converse and contrapositive of the statement: ‘’If there is vacation then i will go to Goa'
Show that the set {-i,i,1,-1} is the abelian group with usual multiplication. Also, show that it is cyclic group.
Define a ring. Show that set of all 3×3 matrices with real entries forms a ring with usual matrix addition and matrix multiplication.
Let (Z,+) be a group. Consider the set H= {5n: n is an integer}. show that H is subgroup of Z. Find the all cosets of H in Z.
There are two shopping malls next to each other. A with the sign board as “Good items are not cheap' and B with sign board as “Cheap items are not good”. Covert this into symbolic proposition. Do they mean same?
Define the identity element of a group. For the set of integer a binary operation is defined by x*y=x+y-2 . Find the identity element and inverse of each integer.
Define cycle in a graph. For the given tree find the pre order post order and in order traversal.
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Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Discrete Mathematics (DM) (Summer 2026, B.E. · Computer Engineering, Sem 4). Features complete 70-mark regular & remedial examination pattern, official marking distribution across all 5 questions, and direct 1-click official PDF download.
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