Questions: Determine whether ⊂, ⊆, both, or neither can be placed in the blank to make the following true. prosciutto, olives, kumquats, liver, mushrooms, quail mushrooms, kumquats, liver, prosciutto, quail

Determine whether ⊂, ⊆, both, or neither can be placed in the blank to make the following true.
prosciutto, olives, kumquats, liver, mushrooms, quail mushrooms, kumquats, liver, prosciutto, quail

Transcript text: Determine whether $\subset, \subseteq$, both, or neither can be placed in the blank to make the following true. \{prosciutto, olives, kumquats, liver, mushrooms, quail\} $\qquad$ \{mushrooms, kumquats, liver, prosciutto, quail\}

Solution

Solution Steps

Step 1: Understand the symbols

The symbol $ \subset $ represents a proper subset, meaning every element of the first set is in the second set, but the second set has at least one additional element.
The symbol $ \subseteq $ represents a subset, meaning every element of the first set is in the second set, and the two sets could be equal.

Step 2: Compare the sets

Let $ A = \{ \text{prosciutto, olives, kumquats, liver, mushrooms, quail} \} $.
Let $ B = \{ \text{mushrooms, kumquats, liver, prosciutto, quail} \} $.
Observe that $ A $ contains the element "olives," which is not in $ B $.

Step 3: Determine the relationship

Since $ A $ has an element not in $ B $, $ A $ is not a subset of $ B $. Therefore, neither $ \subset $ nor $ \subseteq $ can be placed in the blank to make the statement true.
The correct answer is neither.

Final Answer

$\boxed{\text{neither}}$

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