Set Membership and Elements

The governing concept is cardinality: count the distinct values that satisfy both conditions simultaneously. The odd decimal digits are 1, 3, 5, 7, and 9. Applying x > 2 removes 1 because it is not greater than 2, leaving F = {3, 5, 7, 9}. This set has four distinct elements, so option B is correct. The answer is not 5 because that counts all odd digits without applying the inequality. It is not 3 because one of the valid digits has been missed, and it is not 6 because 6 is even and cannot satisfy the first condition. The word “digit” restricts the universe to the ten decimal digits from 0 through 9.

The central concept is set membership. The symbol ∈ means “is an element of,” so the statement is true when the object on the left is explicitly listed inside the set on the right. Since r appears in M = {p, q, r, s}, the statement r ∈ M is true, making option A correct. The symbol t is not listed in M, so t ∈ M is false. The expression M ∈ r is not supported by the information given and reverses the usual membership relationship; r has not been defined as a set containing M. Finally, q is listed in M, so q ∉ M is false. Thus exactly one option is valid.

The governing concept is membership in a roster-form set. A number belongs to N only when it is explicitly listed between the braces. The set N contains exactly three elements: 5, 10, and 15. Therefore 5 ∈ N, 10 ∈ N, and 15 ∈ N are true statements. The number 20 is not listed, so 20 ∉ N and the statement 20 ∈ N is false. It does not matter that 20 is also a multiple of 5; the displayed set is finite and does not automatically include every number following a visible pattern. Option A, B, and C are therefore true, while option D is the only false statement and is the required answer.

The governing concept is membership in a set defined by a condition. To identify the member, solve the equation 2x = 10 by dividing both sides by 2. This gives x = 5. The value 5 is a natural number, and substitution confirms the condition because 2 × 5 = 10. Therefore 5 belongs to X. The other choices do not satisfy the defining equation: 2 × 2 = 4, 2 × 10 = 20, and 2 × 20 = 40. Membership is determined by satisfying every part of the set-builder description, not merely by being related to a number appearing in the equation. Hence option A is the only correct answer.