ACSL Elementary - DeMorgan's Law

A. Notes

DeMorgan’s Law tells us how to negate a combined statement such as A OR B, and A AND B.

NOT (A OR B) = NOT A AND NOT B i.e. ~ ( A + B ) = ~A * ~B

NOT (A AND B) = NOT A OR NOT B i.e. ~( A * B) = ~A + ~B

Here, A and B can be any statement such as "Noah eats more apples than Justin" or "x < 3"

B. HW Practice

1. Let A be " x<3" and B be " x >10" , draw a number line, color all numbers less than 3 red, and all number greater than 10 blue, and the rest yellow. What is the color of a number that is A OR B? What is the color of a number that is NOT(A OR B)? What is the color of a number that is "NOT A AND NOT B"?

2. Let A be " x<3" and B be " x >10" , draw a number line, color all numbers less than 3 red, and all number greater than 10 blue, and the rest yellow. Can you find a number that is A AND B? Can you find a number that is NOT(A AND B)? Can you find a number is "NOT A OR NOT B"?

3. Can you think of another example similar to the above to prove DeMorgan's Law? For instance, you can assume that A is " y > 0" and B is " y < 5". What do ~ ( A + B ) and ~A * ~B mean then? Are they always equivalent? What do ~( A * B) and ~A + ~B mean? Are they always equivalent?

4. Simply the following:

~ (A + B) + ~(A + ~B)

~ (A * B) + ~(A * ~B)