if we consider negative integers of set S, then the E option goes out, am i right? for example 3^-1=1/3

OE


Set S is the infinite set of all odd integers: {… −7, −5, −3, −1, 1, 3, 5, 7 …}. Adding two odd integers yields an even integer, so (A) is not correct. For example, 5 + 3 = 8. Subtracting two odd integers yields an even integer, so (B) is not correct. For example, 5 − 3 = 2. (C) is correct because multiplying any two odd integers always results in another odd integer—for example, 3 × −9 = −27, and 7 × 11 = 77. This is because when neither number has 2 as a factor, the product will not have 2 as a factor (since 2 is a prime number). (D) is not correct, because dividing two elements from Set S will often produce a non-integer value. For example, 5 ÷ 3 ≈1.67. Finally, (E) is correct, because raising an odd number to any power will yield an odd result for basically the same reasons as (C) above. Repeated multiplication by the same odd number continues to yield products that lack 2 as a factor and are thus, by definition, odd. For example, \(3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81\). Answer is (C) and (E).

Thanks for the explanation. but still I do not know why did not consider 3^-1 or -5^3 which make the option E incorrect, please explain on this.

You're correct, mohammadmh91

Since the question doesn't restrict the values of a and b to POSITIVE integers only, answer choice E is incorrect.

So, for example (as you already suggested), if \(a = 3\) and \(b = -1\), then \(a^b = 3^{-1} = \frac{1}{3}\), which is not an odd integer.

Actual answer: C only

Thanks for the clarification.