a) Claim. If and thenProof. Assume and Then there exists a function fsuch that Since Therefore, (b) Claim. If and thenProof. Suppose Then B is a proper subset of C. ThusThis implies But and,since B and are disjoint, By hypothesis,Thus a contradiction.(c) Claim. If and thenProof. Assume andCase 1. Then by substitution inCase 2. Then by transitivity, (d) Claim. If and then there exists a functionProof. Assume Then there exists a functionSince g is one-to-one, every b in B has exactly one pre-image in A. Thusthe set is the pre-image of b under g} is a function. Thisfunction is onto A, because for each a in A, and so

Stat notes week 13 Confidence intervals for the mean and proportions Suppose X is a random variable with known standard deviation σ= 1.7. Construct a two sided symmetric 95% confidence interval using the sample mean X = 8.5 from n=25 samples - X -ε