JAMIA 2025 — Mathematics PYQ

Statement S1 Analysis

S1: There exists infinite sets $A,B,C$ such that $A\cup C=B\cup C$ is finite.

To check if this is possible, let's look at the properties of the Union of sets.

If a set $X$ is a subset of $Y$ (i.e., $X\subseteq Y$), and $Y$ is finite, then $X$ must also be finite.

In the expression $A\cup C$, both sets $A$ and $C$ are subsets of the result:

• If $A\cup C$ is finite, then every subset of $A\cup C$ (including $A$ and $C$) must be finite.

• Therefore, it is mathematically impossible for $A,B,$ or $C$ to be infinite if their union is finite.

Conclusion for S1: Statement S1 is False.

Statement S2 Analysis

S2: There exists two irrational numbers $x$ and $y$ such that $(x+y)$ is rational.

We can prove this by providing a simple counter-example (or "existence" example):

Let $x=\sqrt{2}$ (which is irrational).

Let $y=-\sqrt{2}$ (which is also irrational).

Now, let's calculate their sum:

Since $0$ is a rational number (it can be written as $0/1$), the statement is true.

Another example:

Let $x=3+\sqrt{5}$ and $y=3-\sqrt{5}$. Both are irrational, but:

Since $6$ is rational, the statement holds.

Conclusion for S2: Statement S2 is True.

Final Comparison

Statement	Status	Reason
S1	False	Union of infinite sets is always infinite.
S2	True	Irrational parts can cancel out during addition.

Final Answer:

S1 is False and S2 is True.