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without using truth table prove that

[(p˅q)˄(p→r)˄(q→r)]→r is a tautology.

[(p˅q)˄(p→r)˄(q→r)]→r is a tautology.

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without using truth table prove that

¬(p↔q) and ¬p↔q are logically equivalent.

¬(p↔q) and ¬p↔q are logically equivalent.

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Write a sequence formula?????

1,4,9,16,25,.........

1,4,9,16,25,.........

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Use Mathematical Induction to prove the following:

(i) 1

2 + 22 + 32 + … + n

2 =

(i) 1

2 + 22 + 32 + … + n

2 =

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Q. where the concept of a function first arose? Also find that who has arosed this concept? Also describe How this concept was first used?

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who has developed the concept of congurences and when he developed the concept?

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Who has developed the concept of congurences and when has he developed the concept?

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A relation R defined on a set S is said to be ………… if b R a

∀a,b∈S

.

a. transitive

b. reflexive

c. symmetric

d. none of the above

∀a,b∈S

.

a. transitive

b. reflexive

c. symmetric

d. none of the above

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For each of the following words: if it is a sentence in the grammar, draw its derivation

tree; if it is not a sentence, explain carefully how you know that it is not a sentence.

(i) a/b − c/d ∗ e + f

tree; if it is not a sentence, explain carefully how you know that it is not a sentence.

(i) a/b − c/d ∗ e + f

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identity , equal , equivalent --- are used in mathematics . are identity and equal both same ?? if same then we can put = sign in every identity . but Identities are sometimes indicated by the triple bar symbol ≡ instead of = ------ what does it mean?? if we use the triple bar ≡ , then it becomes equivalent .now we can say identities are sometimes equivalent. is it true? it becomes confusing .please explain with example . please help to learn the actual concept.

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