Analogical Reasoning Math Example 4

Follow the full solution, then compare it with the other examples linked below.

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In arithmetic, the number

0

is the identity for addition (

a+0=a

). By analogy, identify the identity element for set union and for set intersection.

1
Union identity: $A \cup \emptyset = A$ for every set $A$ . The empty set $\emptyset$ plays the role of $0$ in addition.
2
Intersection identity: $A \cap U = A$ for every set $A$ (where $U$ is the universal set). The universal set $U$ plays the role of $1$ in multiplication.
3
So: $\emptyset$ is the additive (union) identity; $U$ is the multiplicative (intersection) identity.

Answer

A \cup \emptyset = A,\quad A \cap U = A

Identities in set algebra correspond to identity elements in arithmetic. The analogy

\cup \leftrightarrow +

and

\cap \leftrightarrow \times

maps

0 \leftrightarrow \emptyset

and

1 \leftrightarrow U

About Analogical Reasoning

Drawing conclusions about a new situation by recognizing its structural similarity to a better-understood situation.

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