Pro0. We show 2∈A∩(BUC) implies z∈(AnB)U( AnC) and vice-versa First, we show z∈A∩(BUO) implies z∈(AnB)∪(AnC). Assume z∈An(BUC) Then z is in A and z is also in B or C. Thus, z is in either AnB or Anc, which implies z∈(A∩B)∪(A∩C) Now, we show z∈(AnB)u(AnC) ) implies z∈An(BUC). Assume z∈(A∩B)∪(AnC) Then z is in both A and b or else z is in both a and C. Thus, z is in a and z is also in b or C. This implies z∈A∩(BUC) 4.2 Method #2: Construct a Chain of lffs In order to prove that P is true if and only if Q is true 1. Write, We construct a chain of if-and-only-if implications 2. Prove p is lent to a second statement which is equivalent to a third staement and so forth until you reach Q This method is generally more difficult than the first, but the result can be a short, elegant proof Example The standard deviation of a sequence of values T1, 2, .. n is defined to be √(x1-p)2+(x1-p)2+…+(xn-p)2 where u is the average of the values: Theorem 5. The standard deviation of a sequence of values 1, .. n is zero if and only if all th values are equal to the mean For example, the standard deviation of test scores is zero if and only if everyone scored exactly the class average Proof. We construct a chain of"if and only if"implications. The standard deviation of C1,...,In is zero if and only if √(x1-1)2+(x1-p)2+…+(x1n-p)2 where u is the average of 1, .. In. This equation holds if and only if (x1-1)2+(x1-1)2+…+(xn-p)2=0 since zero is the only number whose square root is zero. Every term in this equation is nonnegative, so this equation holds if and only every term is actually 0. But this is true if and only if every value ri is equal to the mean u� � 10 Proofs Proof. We show z ∈ A ∩ (B ∪ C) implies z ∈ (A ∩ B) ∪ (A ∩ C) and vice­versa. First, we show z ∈ A ∩(B ∪ C) implies z ∈ (A ∩ B)∪(A ∩ C). Assume z ∈ A ∩(B ∪ C). Then z is in A and z is also in B or C. Thus, z is in either A ∩ B or A ∩ C, which implies z ∈ (A ∩ B) ∪ (A ∩ C). Now, we show z ∈ (A∩B)∪(A∩C) implies z ∈ A∩(B∪C). Assume z ∈ (A∩B)∪(A∩C). Then z is in both A and B or else z is in both A and C. Thus, z is in A and z is also in B or C. This implies z ∈ A ∩ (B ∪ C). 4.2 Method #2: Construct a Chain of Iffs In order to prove that P is true if and only if Q is true: 1. Write, “We construct a chain of if­and­only­if implications.” 2. Prove P is equivalent to a second statement which is equivalent to a third staement and so forth until you reach Q. This method is generally more difficult than the first, but the result can be a short, elegant proof. Example The standard deviation of a sequence of values x1, x2, . . . , xn is defined to be: 2 (x1 − µ)2 + (x1 − µ)2 + . . . + (xn − µ) where µ is the average of the values: x1 + x2 + . . . + xn µ = n Theorem 5. The standard deviation of a sequence of values x1, . . . , xn is zero if and only if all the values are equal to the mean. For example, the standard deviation of test scores is zero if and only if everyone scored exactly the class average. Proof. We construct a chain of “if and only if” implications. The standard deviation of x1, . . . , xn is zero if and only if: (x1 − µ)2 + (x1 − µ)2 + . . . + (xn − µ)2 = 0 where µ is the average of x1, . . . , xn. This equation holds if and only if 2 (x1 − µ) 2 + (x1 − µ) + . . . + (xn − µ) 2 = 0 since zero is the only number whose square root is zero. Every term in this equation is nonnegative, so this equation holds if and only every term is actually 0. But this is true if and only if every value xi is equal to the mean µ