Consider the following reduction ad absurdum proof in propositional logic. the justification lines (to the right) are missing. fill in the blanks. 1. a ⊃ ~b premise 2. ~b ⊃ (c . b) premise 3. ~c v ~b premise 4. a premise / show x 5. ~x assumption for purposes of indirect proof 6. ~b 7. ~(c . b) 8. b 9. ~a 10. a . ~a 11. ~x ⊃ (a . ~a) 12. ~ (a . ~a) 13. x 4,2 mp; 3 dm; 7,2 mt; 8,1 mt; 4, 9 conj.; 6-11 indirect proof; nc; 5,13 mt 3,1 mp; 3 dm; 6,2 mt; 8,1 mp; 4, 9 conj.; 5-11 indirect proof; nc; 12,13 mt 4,1 mp; 3 dm; 7,2 mt; 8,1 mt; 4, 9 conj.; 5-10 indirect proof; nc; 11,12 mt 4,1 mp; 3 dm; 7,2 mt; 7,1 mt; 4, 9 add.; 5-11 indirect proof; conj; 12,1

Jeffery conducted a survey in his school and found that 30 of the 50 eighth grade students' favorite subject is mathematics. based on the results, jeffery concluded that out of the 200 students in his school, 120 students' favorite subject is mathematics. select the statement that is true about jeffery's conclusion. a. jeffery's conclusion is not valid because the sample was biased since only 200 students were surveyed. b. jeffery's conclusion is valid because the sample was random since all of the students were eighth grade students. c. jeffery's conclusion is valid because the sample was random. d. jeffery's conclusion is not valid because the sample was biased since all of the students were eighth grade students.

Diana is painting statues she has 7/8 of a liter of paint each statue requires 1/20 of a liter of paint how many statues can she paint?

Sara has 2 2/3 cup taco meat. she needs 1/6 cup to make one taco. how many tacos can sara make with the taco meat she has?