CSS 220 Sudoku And Python Set Theory

CSS 220 Sudoku And Python Set TheoryPROPOSITIONAL LOGIC: Computer Circuitsp,q,r – input nodess,c – output nodes– XOR gate (the output of an XOR gate is on if and only if its inputs disagree with each other.) ¬ (p ∨ q)– AND gate (the output of an AND gate is on if and only if both of its inputs are on)p q– OR gates (the value of an OR node is on if and only if at least one of its inputs is on)p ∨ q1. Identify which of the following are propositions: (circle one)a. p: Today is Friday PROPOSITION NOT A PROPOSITIONb. p: 3 + 5 = 8 PROPOSITION NOT A PROPOSITIONc. p: Take the quiz PROPOSITION NOT A PROPOSITIONd. p: 7 < 11 PROPOSITION NOT A PROPOSITIONe. p: Put your hat on PROPOSITION NOT A PROPOSITIONf. p: a triangle has 4 sides PROPOSITION NOT A PROPOSITION2. What is the difference between the truth value and the truth table (Semantics).3. NEGATION (NOT): The negation of a proposition can be formed by inserting the word _______ as appropriate. The notation for the negation of p is p.Example:State the negation of the following propositions:a. p: Today is Saturday. p: _______________________________b. p: All mammals respire p: _______________________________c. p: The glass is full p: _______________________________The truth table for a negation is:q¬q4. p in logic corresponds with _________ in set5. CONJUNCTION (AND): A conjunction is formed when two propositions are connected by the word _________.Example:Let p: London is the capital of England.q: Houston is the capital of the United States.State p q: ___________________________________________________The truth table for a conjunction is:pqp q6. Tautology vs Contradiction Examples:7. DISJUNCTION (OR): A disjunction is formed when propositions are joined by the word “or.” Example:Let p: London is the capital of England.q: Houston is the capital of the United States.State p q: ________________________________________________The truth table for a conjunction is:pqp V q8. Populate this truth table for ¬p ∨ ¬qpq¬p¬q¬p ∨ ¬q9. If you have n propositions, how many lines will you have in your truth table?10. Create a truth table for (p q) rpqr(p q)(p q)(p q) r11. IMPLICATION (If-Then): When a proposition p being true implies that another proposition q must also be true then we say that p implies q. p ⇒ qExample:a. If you score 90% or above in this class, then you will get an A.b. My thumb will hurt if I hit it with a hammer.c. x = 2 implies x + 1 = 3.d. If Jimmy loses a tooth, then Jimmy finds a dollar.p (antecedent)q (consequence)p implies qTTTTFFFTTFFT12. EQUIVALENCE (If-And-Only-If): Two propositions p and q are called equivalent statements if each implies the other (p if and only if q): p ≡ q, p↔qp: we will go to the amusement park,q: we will go to the zoopqp ↔ qTTTFFTFF13. Prove that:p ⇒ q ≡ ¬p ∨ qpqp ⇒ q¬ p¬p ∨ q14. Distributive p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (q ∧ r)pq15. De Morgan’s : ¬(p ∨ q) ≡ ¬p ¬qpq16. Simplify: (p ∧ q) ∨ ( ¬q ∧ p)