Which of the following statements are true ? Justify your answer .
(i) The set of dogs is contained in the set of animals.
(ii) The set of all isosceles triangles is contained in the set of all equilateral triangles .
(iii) The set of all rectangles is contained in the set of squares .
(iv) The sets A = {x:x is letters in the word LITTLE} and B= {x:x is a letter in the word TITLE} are equal.
(v) Every set has a proper subset.

Answer

To determine which of the statements are true, we will analyze each statement one by one and provide justifications.

### Step-by-Step Solution:

**(i) The set of dogs is contained in the set of animals.**

- **Justification:** This statement is true. All dogs are animals, meaning that the set of dogs is a subset of the set of animals. In set notation, we can express this as \( \text{Dogs} \subseteq \text{Animals} \).

**(ii) The set of all isosceles triangles is contained in the set of all equilateral triangles.**

- **Justification:** This statement is false. An isosceles triangle has at least two equal sides, while an equilateral triangle has all three sides equal. Therefore, not all isosceles triangles are equilateral. In set notation, we can express this as \( \text{Isosceles} \nsubseteq \text{Equilateral} \).

**(iii) The set of all rectangles is contained in the set of squares.**

- **Justification:** This statement is false. While all squares are rectangles (since they have four right angles), not all rectangles are squares (as rectangles can have unequal adjacent sides). Thus, the set of rectangles is not a subset of the set of squares. In set notation, we can express this as \( \text{Rectangles} \nsubseteq \text{Squares} \).

**(iv) The sets A = {x: x is a letter in the word LITTLE} and B = {x: x is a letter in the word TITLE} are equal.**

- **Justification:** This statement is true. The letters in "LITTLE" are L, I, T, and E, while the letters in "TITLE" are T, I, L, and E. When we consider sets, we ignore duplicates. Thus, both sets can be expressed as \( A = \{L, I, T, E\} \) and \( B = \{L, I, T, E\} \), making them equal. In set notation, we can express this as \( A = B \).

**(v) Every set has a proper subset.**

- **Justification:** This statement is false. A proper subset is defined as a subset that is not equal to the original set. For a set with only one element (e.g., \( \{a\} \)), the only subset is itself, which cannot be a proper subset. Therefore, not every set has a proper subset.

### Summary of Truth Values:
- (i) True
- (ii) False
- (iii) False
- (iv) True
- (v) False