Difference between revisions of "2024 AMC 10A Problems/Problem 14"

Revision as of 12:38, 11 August 2024

Since you came this far already, here's a math problem for you to try: What is 9+10? (A) 19 (B) 20 (C) 21 (D) 22 (E) 23

Define = satisfying the following axioms $a=a$

$a=b \implies b=a$

$a=b, b=c \implies a=c$

Define $\mathbb{N}$

$0 = \emptyset = \{ \}$

$0 \in \mathbb{N}_0$

(note we use $\mathbb{N}_0$ cause I'm one of those $0 \notin \mathbb{N}$ people)

$S(n) := n \cup \{n \}$

$n \in \mathbb{N}_0 \implies S(n) \in \mathbb{N}_0$

$\forall n \in \mathbb{N}_0, n \not= 0, \exists m \in \mathbb{N}_0 : S(m)=n$

@@ Line 5: / Line 5: @@
 ==Solution 1==
 Define = satisfying the following axioms
-<math>a=a</math>\\
+<math>a=a</math>
-<math>a=b \implies b=a</math>\\
-<math>a=b, b=c \implies a=c</math>\\
+<math>a=b \implies b=a</math>
-Define <math>\mathbb{N}</math>\\
-<math>0 = \emptyset = \{ \}</math>\\
+<math>a=b, b=c \implies a=c</math>
-<math>0 \in \mathbb{N}_0</math>\\
-(note we use <math>\mathbb{N}_0</math> cause I'm one of those <math>0 \notin \mathbb{N}</math> people)\\
+Define <math>\mathbb{N}</math>
-<math>S(n) := n \cup \{n \}</math>\\
-<math>n \in \mathbb{N}_0 \implies S(n) \in \mathbb{N}_0</math>\|
+<math>0 = \emptyset = \{ \}</math>
-<math>\forall n \in \mathbb{N}_0, n \not= 0, \exists m \in \mathbb{N}_0 : S(m)=n</math>\\
+<math>0 \in \mathbb{N}_0</math>
+(note we use <math>\mathbb{N}_0</math> cause I'm one of those <math>0 \notin \mathbb{N}</math> people)
+<math>S(n) := n \cup \{n \}</math>
+<math>n \in \mathbb{N}_0 \implies S(n) \in \mathbb{N}_0</math>
+<math>\forall n \in \mathbb{N}_0, n \not= 0, \exists m \in \mathbb{N}_0 : S(m)=n</math>