Explanation

Empty set is, in fact, a subset of every set, including itself, because the definition of a subset allows the empty set to be considered a subset of any set.

The intersection of the empty set with any set is always the empty set, making statements about it true.

Explanation

Given b^n = b^m, for b ≠ 0:

Step 1: Analyze the equation

If b^n = b^m, then n = m, or b = 1 or b = -1 with specific conditions on n and m.

Step 2: Consider the case when b ≠ 1 and b ≠ -1

For b ≠ 0, 1, or -1, the equation holds true when n = m.

Step 3: Look for an option that satisfies n = m

Among the given options, one possible solution where n = m is when both n and m are 0, since any non-zero number to the power of 0 is 1.

The answer is: m = 0 and n = 0.

Explanation

Denotative meaning is the literal, dictionary definition of a word.

It contrasts with connotative meaning, which includes emotional or cultural associations.

Explanation

We know the identity:

a³ - b³ = (a - b)(a² + ab + b²)

Here, 8x³ - 27y³ = (2x)³ - (3y)³

= (2x - 3y)((2x)² + (2x)(3y) + (3y)²)

Given 2x - 3y = 8 and xy = 2:

(2x)² + (2x)(3y) + (3y)² = 4x² + 6xy + 9y²

= 4x² + 6(2) + 9y²

= 4x² + 12 + 9y²

We need to express 4x² + 9y² in terms of 2x - 3y and xy:

(2x - 3y)² = 4x² - 12xy + 9y²

64 = 4x² - 12(2) + 9y²

64 = 4x² - 24 + 9y²

4x² + 9y² = 88

Now, 4x² + 6xy + 9y² = 88 + 12 = 100

So, 8x³ - 27y³ = (2x - 3y)((2x)² + (2x)(3y) + (3y)²)

= 8 × 100

= 800

Explanation

The Halting problem is about determining whether a given program will terminate (halt) or run indefinitely for a specific input.

It was proven by Alan Turing that there is no general algorithm to solve this problem for all possible programs.

Explanation

Over 6000 languages are spoken globally, including many with distinct dialects and regional variations.

This linguistic diversity is mainly concentrated in Asia and Africa, where countries like Papua New Guinea alone have over 800 languages.

Explanation

Expansion of metal on heating is a physical change because it does not alter the metal’s chemical composition.

When cooled, the metal returns to its original size, showing that the change is reversible.

Explanation

Explanation

"The" is used for a specific person or thing known to the speaker and listener.

"Boy in red shirt" is a specific boy, so "The" is appropriate.

Explanation

To find the fourth proportional, let's denote it as x. The proportion will be:

3 : 9 = 7 : x

We can cross-multiply:

3x = 9 × 7

3x = 63

x = 63 / 3

x = 21