B. Tech 2025 — Question Paper

Q1

Ever wonder how we can neatly sort all integers into distinct families? Let's look at a relation where two integers, x and y, are connected if their sum is an even number.

Let's decode this rule before doing any heavy lifting. For the sum of two integers to be even, they must share the same 'parity'. That means they are either both even, or both odd.

To classify this relation, we test three properties. First: is it reflexive? Meaning, does every integer relate to itself?

Well, x plus x is just 2x. Since two times any integer is always a multiple of two, it's always even. So yes, the relation is perfectly reflexive.

But what happens if we swap the numbers? That's our second test: symmetry. If x relates to y, does y relate to x?

Because addition works the same in any order, x plus y is exactly the same as y plus x. If one is even, the other must be too. Symmetry checks out.

Finally, the trickiest one: transitivity. If x relates to y, and y relates to a third number z, does x relate directly to z?

Let's picture it. If x and y have an even sum, they share the same parity. And if y and z have an even sum, they also share the same parity.

This means x and z are forced to have that exact same parity. Since they match, their sum is even, making the relation transitive.

A relation that is reflexive, symmetric, and transitive has a special name. It's an equivalence relation, which perfectly matches our fourth option.