Math 61 Midterm 1 Review

ucla | MATH 61 | 2022-10-17T11:03

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Table of Contents

• Recall
• Notes
• Lecture

Recall

Notes

Supplemental Definitions

• vacuously
  • a statement of P→Q is true if P is false

Example Problems

• Relations

  • $X={\mathbb{Z}}_{>0}$ R on X s.t. $R={(2n,2n+1)

    n\in\mathbb Z}$

  • show antisymmetric? transitive?
    • antisymmetric
      • R is antisymmetric if when xRy and yRx, x=y for x,y $\in$ X
      • (x,y) $\in$ R if x is even, and y is odd, so (y,x) $\notin$ R
      • $\therefore$ R is antisymmetric vacuously
• Bijection proof
  • A is the set of subsets (S) of the Power set of Y
  • B is the set of subsets (T) of the Power set of X
  • X and {y} partitions Y
  • prove bijection A→B, S→S-{y}
  • prove injective:
    • f is injective if $f(A_1)=f(A_2),A_1,A_2\in A⟹A_1=A_2$
    • sps. $f(A_1)=f(A_2)=A_1-y=A_2-y$
    • then, $A_1=A_1-y\cup y$ and $A_2=A_2-y\cup y\therefore$ f is injective b/c A1=A2
  • prove surjective
    • if for each B in B, f(A1)=B,
    • so A1=B1 union {y}

Lecture

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**SUMMARY
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