Sets - Equality of Sets-advanced

Let $A = \{x : x \text{ is a letter in the word 'FOLLOW'}\}$ and $B = \{y : y \text{ is a letter in the word 'WOLF'}\}$. Are $A$ and $B$ equal?

$A = \{F, O, L, W\}$ $B = \{W, O, L, F\}$ Since every element of $A$ is in $B$ and every element of $B$ is in $A$, $A = B$.

In set notation, we ignore the repetition of letters like 'L' and 'O' in 'FOLLOW'. The distinct elements are the same for both words, so the sets are equal regardless of order.

Given $X = \{x : x^2 - 5x + 6 = 0\}$ and $Y = \{2, 3\}$, prove $X = Y$.

First, solve the quadratic equation for $X$: $x^2 - 5x + 6 = 0$ $(x - 2)(x - 3) = 0$ $x = 2 \text{ or } x = 3$ So, $X = \{2, 3\}$. Since $Y = \{2, 3\}$, it follows that $X = Y$.

By solving the defining property of set $X$, we find its elements are exactly the same as those listed in set $Y$.

If $P = \{n : n \in \mathbb{N}, n < 4\}$ and $Q = \{1, 2, 3, 3\}$, are $P$ and $Q$ equal? Also, find if they are equivalent.

For set $P$: $P = \{1, 2, 3\}$ For set $Q$: $Q = \{1, 2, 3\}$ (after removing duplicates) Since $P$ and $Q$ contain the same elements, $P = Q$. Cardinality: $n(P) = 3$ $n(Q) = 3$ Since $n(P) = n(Q)$, they are also equivalent.

Sets are equal because they share the same unique elements. They are equivalent because their count of distinct elements is identical.