Math 61 Midterm 1 Review

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

Table of Contents

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 $f(A_1)=f(A_2), A_1,A_2\in A\implies A_1=A_2$
      • 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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