Topic 2: Logic Consolidation Worksheet

1. Know your statements! Use the following statement: If two angles are vertical angles, then they are congruent. converse If two angles are congruent, then they are vertical angles inverse If two angles are not vertical angles, then they are not congruent. contrapositive If two angles are not congruent, then they are not vertical angles. biconditional Two angles are vertical angles if and only if they are congruent. 2. Know your symbols! Use the following statement: If two lines are perpendicular, then they form a right angle. p two lines are perpendicular ~q they do not form a right angle p → q If two lines are perpendicular, then they form a right angle. q → p If two lines form a right angle, then they are perpendicular. ~p → ~q If two lines are not perpendicular, then they do not form a right angle. ~q → ~p If two lines do not form a right angle, then they are not perpendicular. p ↔ q Two lines are perpendicular if and only if they form a right angle. ~q → p If two lines do not form a right angle, then they are perpendicular. q → ~p If two lines form a right angle, then they are not perpendicular. 3.1 Determine whether the following arguments are valid or invalid. Drag and drop them into the appropriate box. Valid If you live in San Francisco, then you live in California. Maria lives in San Francisco. Therefore, she lives in California. All mammals are warm blooded. Bats are mammals. Therefore, they are warm blooded. If the stereo is on, then the volume is loud. If the volume is loud, then the neighbors will complain. Therefore, if the stereo is on then the neighbors will complain. If the sun is shining, it's a beautiful day. If it's a beautiful day, we will have a picnic. Therefore, if the sun is shining, we will have a picnic. If you run fast, you will win the race. Mark did not win the race. Therefore, he did not run fast. If Ryan gets an A on the final, he will pass the course. Ryan did not pass the course. Therefore, he did not get an A on the final. If I am hungry, then I will eat lunch. If I eat lunch, then I will have a sandwich. Therefore, if I didn't have a sandwich, then I wasn't hungry. Invalid If the car is running, then the key is in the ignition. The key is in the ignition. Therefore, the car is running. If you study hard, then you will get a good grade. Max did not study hard. Therefore, he did not get a good grade. If Wayne reads the book by Friday, he will write the report on Saturday. Wayne did not read the book by Friday. Therefore, he did not write the report on Saturday. If the hill is covered with snow, then I will go sledding. I went sledding. Therefore, the hill was covered with snow. If I go to the mall, then I will see my friends. If I go to the mall, then I will spend money. Therefore, if I see my friends then I will spend money. 3.2 Negate the following statement:“100 is a multiple of 10 or 5.” 100 is not a multiple of 10 or 5 100 is not a multiple of 10 and not a multiple of 5 100 is not a multiple of 10 or not a multiple of 5 4. Basic Truth Tables Match the following tables to the correct proposition statements Disjunction "OR" Conjunction "AND" Negation "NOT" Implication "IF, THEN" Bi-conditional "IF AND ONLY IF" 5. Given p and q are false, and r and s are true statements, what is thetruth value of (~p ⋁ s) ↔ ~(q ⋀ ~r). (~p ⋁ s) ↔ ~(q ⋀ ~r)(~F ⋁ T) ↔ ~(F ⋀ ~T)(T ⋁ T) ↔ ~(F ⋀ F)(T) ↔ ~(F)(T) ↔ (T)T 6. Bi-conditional truth Table The bi-conditional statement ‘if and only if’ is the conjunction of if p then q and if q then p. Complete the table below to prove that:(p → q) ∧ (q → p) ≡ p ⟺ q p q p↔q p→q q→p (p → q) ∧ (q → p) T T T T T T T F F F T F F T F T F F F F T T T T 7. Equivalence Prove that the following statement is equivalent by filling in the truth tables~(p ∨ q) ≡ ~p ∧ ~q p q ~p ~q p ∨ q ~(p ∨ q) ~p ∧ ~q T T F F T F F T F F T T F F F T T F T F F F F T T F T T