Let's analyze each of the statements for sets [tex]\( A = \{1, 2, 3, \ldots, 100\} \)[/tex] and [tex]\( B = \{2, 4, 6, \ldots, 200\} \)[/tex] to determine the appropriate quantifier (either [tex]\(\forall\)[/tex] or [tex]\(\exists\)[/tex]) to fill in the blanks.
Statement a) [tex]\( \quad \_ \quad x \in A , \ x \in B \)[/tex]:
We need to determine if there exists an element [tex]\( x \)[/tex] that is in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. Set [tex]\( A \)[/tex] contains elements from 1 to 100, and set [tex]\( B \)[/tex] contains even elements from 2 to 200. There is at least one common element (e.g., 2, 4, 6, etc.), so we use the existential quantifier [tex]\(\exists\)[/tex]:
[tex]\[ \exists x \in A, \ x \in B \][/tex]
Statement b) [tex]\( \quad \_ \quad x \in B , \ x \in A \)[/tex]:
Here, we need to determine if there exists an element [tex]\( x \)[/tex] in set [tex]\( B \)[/tex] that is also in set [tex]\( A \)[/tex]. As we found in statement (a), there are several elements in set [tex]\( B \)[/tex] that are also in set [tex]\( A \)[/tex]. Hence, we use the existential quantifier [tex]\(\exists\)[/tex]:
[tex]\[ \exists x \in B, \ x \in A \][/tex]
Statement c) [tex]\( \quad \_ \quad x \in A , \ x \ \text{is even}\)[/tex]:
We need to determine if all elements [tex]\( x \)[/tex] in set [tex]\( A \)[/tex] are even. Set [tex]\( A \)[/tex] includes both odd and even numbers from 1 to 100. Since not all elements are even, we use the existential quantifier [tex]\(\exists\)[/tex]:
[tex]\[ \exists x \in A, \ x \ \text{is even} \][/tex]
Statement d) [tex]\( \quad \_ \quad x \in B , \ x \ \text{is even}\)[/tex]:
We need to determine if all elements [tex]\( x \)[/tex] in set [tex]\( B \)[/tex] are even. Since set [tex]\( B \)[/tex] is defined as containing even numbers from 2 to 200, all elements in [tex]\( B \)[/tex] are indeed even. Therefore, we use the universal quantifier [tex]\(\forall\)[/tex]:
[tex]\[ \forall x \in B, \ x \ \text{is even} \][/tex]
Thus, the answers to fill in the blanks are:
a) [tex]\( \exists \)[/tex]
b) [tex]\( \exists \)[/tex]
c) [tex]\( \exists \)[/tex]
d) [tex]\( \forall \)[/tex]
