Analogical Reasoning Math Example 2

Solution

1. 1
   Analogy: addition $\leftrightarrow$ union $\cup$; multiplication $\leftrightarrow$ intersection $\cap$.
2. 2
   Commutative law holds in both: $a+b=b+a \leftrightarrow A\cup B=B\cup A$ (and similarly for $\cap$).
3. 3
   Distributive law holds: $a(b+c)=ab+ac \leftrightarrow A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$.
4. 4
   Cancellation fails: in arithmetic, $a+c=b+c \Rightarrow a=b$. But $A\cup C=B\cup C$ does not imply $A=B$ (e.g., if $A\subset C$ and $B\subset C$, then $A\cup C=B\cup C=C$ for any $A,B\subseteq C$).