Given: p is true
prove: p → q is true
assume p and ~q are both true. ~q → r, and r → ~p. therefore, ~p and p cannot be true, so p and ~q cannot be true. therefore, p → q is true.

what type of proof is illustrated above?

a.
proof by contradiction
b.
proof by contraposition
c.
proof by law of detachment
d.
proof by theorem

Simplify 5 square root of 7 end root plus 12 square root of 6 end root minus 10 square root of 7 end root minus 5 square root of 6 . (1 point) 5 square root of 14 end root minus 7 square root of 12 5 square root of 7 end root minus 7 square root of 6 7 square root of 12 end root minus 5 square root of 14 7 square root of 6 end root minus 5 square root of 7

Billy plotted −3 4 and −1 4 on a number line to determine that −3 4 is smaller than −1 4 is he correct? explain why or why not?

Astudent is determining the influence of different types of food on the growth rate of spiders. she feeds one group of spiders two caterpillars twice a week. the other group received similarly-sized beetles twice a week. she records the amount of time required for individuals to reach sexual maturity. her hypothesis is that the spiders feeding on caterpillars will have a faster growth rate because caterpillars have a higher protein content compared to beetles. in this experiment, what is the dependent variable? spiders assigned to different feeding groups. time required to reach sexual maturity number of prey items fed twice a week. growth rates of each group.