Consider the following statements: S1: There exists infinite sets A, B, C such…

Analysis of Statement S1:

“There exists infinite sets A, B, C such that A ∩ (B ∪ C) is finite.”

To prove an existential statement ("there exists"), we only need to construct one working scenario. Let's define three infinite sets:

• Let A = {x | x is a negative integer} = {-1, -2, -3, -4, .....}

• Let B = {x | x is a positive even integer} = {2, 4, 6, 8, .......}

• Let C = {x | x is a positive odd integer} = {1, 3, 5, 7, .......}

All three sets (A, B, and C) are completely infinite.

Now let's compute the operations:

1. First, find (B ∩ C): This combines all positive evens and odds, giving us the set of all positive integers: {1, 2, 3, 4, ........}.

2. Next, find A ∩ (B ∪ C): We look for elements common to the negative integers (A) and the positive integers (B ∪ C). There are none.

A ∩ (B ∪ C) = ϕ

The empty set (emptyset) has a size of 0, which is finite. Since we found a case where the result is finite, S1 is correct.

Analysis of Statement S2:

“There exists two irrational numbers x and y such that (x+y) is rational.”

An irrational number is a number that cannot be written as a simple fraction. Let's pick two well-known irrational numbers designed to cancel out their irrational parts when added together:

• Let x = 2 + √3 (This is irrational)

• Let y = 2 - √3 (This is also irrational)

Now, let's find their sum (x + y):

(2 + √3) + (2 - √3) = 2 + 2 + √3 - √3 = 4

The number 4 can be written as 4/1, which means it is a rational number.

Since we successfully found a pair of irrational numbers whose sum is rational, S2 is correct.