Session 3B
Set identities with element-chasing
45-75 min - work through lesson notes, practice, and the MCQ check.
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Session tips
- Finish the MCQ before marking complete.
- Check the answer outlines after trying on paper.
- Mark complete to update your certificate progress.
Lesson notes
- Set identities like A intersect (B union C) = (A intersect B) union (A intersect C) are proven by showing both directions using element-chasing. For the "subseteq" direction, assume x is in the left side, translate that into logical form, then deduce x satisfies the right-side condition. For the "supseteq" direction, assume x is in the right side and reverse the logic. This is similar to proving "iff" statements because equality of sets is equivalent to "x is in the left iff x is in the right."
- De Morgan's laws for sets match De Morgan's laws for logic. For example, the complement of a union is the intersection of complements, and the complement of an intersection is the union of complements. For difference, A \ (B union C) equals (A \ B) intersect (A \ C). These identities become almost automatic if you always translate set membership into "and/or/not." The deeper skill here is learning to move fluently between set notation and logic, because many later topics (probability, analysis, abstract algebra) use the same pattern.
Practice
- 1Prove: A intersect (B union C) = (A intersect B) union (A intersect C)
- 2Prove: A \ (B union C) = (A \ B) intersect (A \ C)
Show answers / outlines
Answers
- 1Show each side is a subset of the other using element-chasing.
MCQ
To prove A intersect (B union C) = (A intersect B) union (A intersect C), you must: