Explanation

Ext(A) represents the exterior of set A, which is the set of all points in the universal set X that are not in the closure of A (denoted by A°).

The interior of the closure of A (Int(A°)) consists of all points in A° that are not boundary points.

Since the exterior of A consists of all points in X that are not in A°, Ext(A) is equal to Int(A°).

Explanation

Brave means بہادر

Fearless means بے خوف

Explanation

The given inequality is 3 < x < 7, which can be rewritten as |x-5|<2.

Explanation

Perplexed means پریشان

Puzzle means پریشان

Explanation

Candid means صاف گوئی

Secretive means خفیہ

Explanation

• A convergent sequence in a metric space is bounded.
• This means that the sequence has a finite limit, and all terms of the sequence are within a finite distance from this limit.
• As a result, the sequence is bounded, meaning it has a finite upper and lower bound.

Explanation

• Every subgroup of a cyclic group is also cyclic.
• This is because a subgroup of a cyclic group is generated by some power of the generator of the original group.

Explanation

The correct answer is: ∑1/n^p converges if p > 1.

The p-series ∑1/n^p is a well-known series in mathematics, and its convergence properties are as follows:

• If p > 1, the series converges.
• If p ≤ 1, the series diverges.

This is because the terms of the series decrease slower than the terms of a geometric series with ratio less than 1 when p ≤ 1, and decrease faster than the terms of a geometric series with ratio less than 1 when p > 1.

Explanation

A subset F of a topological space X is closed if and only if F is equal to its closure, denoted by Cl(F).

Explanation

To evaluate the limit, we can start by rewriting the expression as:

lim (sin(2x) + a*sin(x)) / x^3

As x approaches 0, the sine functions can be approximated by their arguments, so we get:

lim (2x + a*x) / x^3 = lim (2 + a) / x^2

For this limit to exist, the numerator must be 0, so we set 2 + a = 0, which gives a = -2.