From the sets given below, select pairs of equal and equivalent sets.
`A={0,1},B={x:x^(2)=25},C={x:x^(2)=x},D={1,2,3,4}.`
`E={3,5,7,11}, F={x:x^(2)=1},G={g,h,i,j},H={3,5,7}`

Answer

To solve the problem of identifying pairs of equal and equivalent sets from the given sets, we will analyze each set step by step.

### Step 1: Identify the elements of each set

1. **Set A**: \( A = \{0, 1\} \)
2. **Set B**: \( B = \{x : x^2 = 25\} \)
- The solutions to \( x^2 = 25 \) are \( x = 5 \) and \( x = -5 \).
- Thus, \( B = \{5, -5\} \).
3. **Set C**: \( C = \{x : x^2 = x\} \)
- Rearranging gives \( x^2 - x = 0 \) which factors to \( x(x - 1) = 0 \).
- The solutions are \( x = 0 \) and \( x = 1 \).
- Thus, \( C = \{0, 1\} \).
4. **Set D**: \( D = \{1, 2, 3, 4\} \)
5. **Set E**: \( E = \{3, 5, 7, 11\} \)
6. **Set F**: \( F = \{x : x^2 = 1\} \)
- The solutions to \( x^2 = 1 \) are \( x = 1 \) and \( x = -1 \).
- Thus, \( F = \{1, -1\} \).
7. **Set G**: \( G = \{g, h, i, j\} \)
8. **Set H**: \( H = \{3, 5, 7\} \)

### Step 2: Compare the sets for equality

Two sets are equal if they contain exactly the same elements.

- **Comparison**:
- \( A = C = \{0, 1\} \) (Equal sets)
- \( B = \{5, -5\} \)
- \( D = \{1, 2, 3, 4\} \)
- \( E = \{3, 5, 7, 11\} \)
- \( F = \{1, -1\} \)
- \( G = \{g, h, i, j\} \)
- \( H = \{3, 5, 7\} \)

Thus, the only pair of equal sets is \( (A, C) \).

### Step 3: Compare the sets for equivalence

Two sets are equivalent if they have the same number of elements.

- **Count of elements**:
- \( |A| = 2 \)
- \( |B| = 2 \)
- \( |C| = 2 \)
- \( |D| = 4 \)
- \( |E| = 4 \)
- \( |F| = 2 \)
- \( |G| = 4 \)
- \( |H| = 3 \)

- **Equivalent sets**:
- \( A, B, C, F \) (all have 2 elements)
- \( D, E, G \) (all have 4 elements)

### Final Result

- **Equal Sets**: \( (A, C) \)
- **Equivalent Sets**: \( (A, B, C, F) \) and \( (D, E, G) \)