Consider the following statement
I. The set of positive powers of 2 is closed under multiplication.
II. The set [1,0,-1] is closed under multiplication.
III. The number 35 has exactly four divisors.
IV. The set [1,0,-1] is closed under addition.
Of the above statement

Options

A. I, II, III are true
B. Only III is true
C. All are false
D. All are true

Answer

To determine the validity of the statements provided, we will analyze each one step by step.

### Statement I: The set of positive powers of 2 is closed under multiplication.

**Step 1:** Define the set of positive powers of 2.
- The set can be represented as {2^1, 2^2, 2^3, ...} = {2, 4, 8, 16, ...}.

**Step 2:** Check closure under multiplication.
- Take any two elements from the set, say 2^m and 2^n (where m, n are positive integers).
- Their product is:
\[ 2^m \times 2^n = 2^{m+n} \]
- Since \(m+n\) is also a positive integer, \(2^{m+n}\) is also in the set of positive powers of 2.

**Conclusion:** The set is closed under multiplication. **(True)**

---

### Statement II: The set {1, 0, -1} is closed under multiplication.

**Step 1:** Define the set.
- The set is {1, 0, -1}.

**Step 2:** Check closure under multiplication.
- Check all possible products:
- \(1 \times 1 = 1\) (in the set)
- \(1 \times 0 = 0\) (in the set)
- \(1 \times -1 = -1\) (in the set)
- \(0 \times 0 = 0\) (in the set)
- \(0 \times -1 = 0\) (in the set)
- \(-1 \times -1 = 1\) (in the set)
- \(-1 \times 0 = 0\) (in the set)

**Conclusion:** All products result in elements that are in the set. **(True)**

---

### Statement III: The number 35 has exactly four divisors.

**Step 1:** Find the divisors of 35.
- The prime factorization of 35 is \(5 \times 7\).

**Step 2:** Calculate the number of divisors.
- The formula for finding the number of divisors from the prime factorization \(p_1^{e_1} \times p_2^{e_2} \) is \((e_1 + 1)(e_2 + 1)\).
- Here, both 5 and 7 are raised to the power of 1, so:
\[ (1 + 1)(1 + 1) = 2 \times 2 = 4 \]
- The divisors of 35 are {1, 5, 7, 35}.

**Conclusion:** The number 35 has exactly four divisors. **(True)**

---

### Statement IV: The set {1, 0, -1} is closed under addition.

**Step 1:** Check closure under addition.
- Check all possible sums:
- \(1 + 1 = 2\) (not in the set)
- \(1 + 0 = 1\) (in the set)
- \(1 + -1 = 0\) (in the set)
- \(0 + 0 = 0\) (in the set)
- \(0 + -1 = -1\) (in the set)
- \(-1 + -1 = -2\) (not in the set)

**Conclusion:** The set is not closed under addition because some sums result in numbers not in the set. **(False)**

---

### Final Evaluation of Statements:
- Statement I: True
- Statement II: True
- Statement III: True
- Statement IV: False

### Summary:
The statements I, II, and III are true, while IV is false.

---